>> Welcome to the Cypress College Math Review on Basic Probability.
Objective one: Simple Probability.
To find the probability of event E, P of E equals the number of ways
that event E can occur, divided by the total number of outcomes in the sample space.
Example one, in a pet store, there are 15 puppies, 22 kittens, and 18 rabbits.
What is the probability of randomly selecting a rabbit?
The probability of rabbit is equal to the number of rabbits, 18,
divided by the total number of pets in the store.
We add 15 plus 22, plus 18 to get 55 total pets.
And as a decimal, this works out to 0.3273.
Example two.
The following data represents the number of classes a student is taking and their gender.
What is the probability that a randomly selected student is taking one class?
The probability of one class, is the number of students taking one class, 48,
and that is the sum of 13 plus 35.
Divided by the total number of students.
This can be found in the bottom right portion of the table, and that's 247.
And that division comes out to 0.1943.
Example three.
What is the probability of drawing a heart from a standard 52 card deck?
Here is an illustration of the 52 cards in the deck.
Notice that all the hearts are in the second row.
The probability of drawing a heart is equal to the number of hearts,
and you can count that there are 13 of them.
Divided by the total number of cards, 52.
Thirteen over 52 reduces to 1/4.
Often when the numbers are small, we leave the answer
in fraction form instead of converting it to a decimal.
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Objective two: Addition Rule for Disjoint Events.
Definition: Two events E and F are disjoint, or mutually exclusive,
if they have no outcomes in common.
The Addition Rule for Disjoint or Mutually Exclusive Events says, "If two events E
and F are disjoint, then the probability of event E or the probability
of event F occurring is given by, P of E or F is equal to P of E plus P of F."
Example four.
What is the probability of drawing a king or a queen form a standard deck of 52 playing cards?
Note that it is not possible for a card to be a queen and a king at the same time.
Therefore, there are no events in common, and the events king and queen are disjoint events.
So, we can use the rule stated above.
Probability of king or queen is equal to probability of king, plus probability of queen.
The probability of drawing a king is 4 over 52, because there are four kings in the deck.
The probability of queen is the same, also 4 over 52.
We add these to get 8 over 52, and this reduces to 2/13 and as a decimal, that is 0.1538.
Example five.
The following data represent the number of classes a student is taking and their gender.
What is the probability that a randomly selected student is taking two classes or three classes?
Notice that these two events are disjoint, because there are no outcomes in common.
That is, there are no students who are taking both two classes
and three classes at the same time.
So, we can use our Addition Rule for Disjoint Events, which says,
"The probability of two classes or three classes, is equal to the probability
of two classes plus the probability of three classes.
The probability of two classes is 64 divided by 247.
The probability for three classes is 94 divided by 247.
This adds up to 158 over 247, and the decimal approximation is 0.6397.
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Objective three: General Addition Rule: If two events are not disjoint,
which means they do have outcomes in common, then you should use the General Addition Rule.
Which says, "For any two events E and F, the probability of event E or the probability
of event F occurring is given by, P or E or F equals P of E,
plus P of F, minus P of E and F. Example six.
What is the probability of drawing a diamond or a king from a standard deck of 52 playing cards?
The probability of diamond of king is equal to the probability of diamond, plus the probability
of king, minus the probability of diamond and king.
Notice that there are 13 diamonds in the deck, there are four kings, and there is one card
which is both a diamond and a king.
Therefore, the probability of diamond is 13 over 52, the probability of king is four over 52,
and the probability of diamond and king is one over 52.
This works out to 16 over 52, which simplifies to 4 over 13.
And as a decimal, that's 0.3077.
Example seven.
Here again is the table showing the number of classes a student is taking and their gender.
What is the probability that a randomly selected student is a female or taking three classes?
Notice that these two events are not disjoint, because they do have outcomes in common.
Meaning, a student can be female and taking three classes at the same time.
So, we must use the General Addition Rule.
Probability of female or three classes equals probability of female plus probability
of three classes, minus probability of female and three classes.
From the table, we see 132 total females, so that probability is 132 over 247.
There are 94 students taking three classes, so that probability is 94 over 247.
And there are 56 students who are both female and taking three classes,
so that probability is 56 over 247.
When we work this out we get 170 over 247, which as a decimal is 0.6883.
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Objective four: Complement Rule.
Definition: Let E be any event, then the complement of E denoted as E C is all
of the outcomes in the sample space that are not outcomes in the event E. An easy way
to understand the idea of complement is that it is the opposite of an event.
For example, if the event is that it is going to rain today, then the complement would be
that it is not going to rain today.
The Complement Rule says that, "If E represents any event and E C represents the complement
of E, then P of E C equals 1 minus P of E."
And we can also say, P of E equals 1 minus P of E C. Example Eight, what is the probability
of not drawing a spade card from a standard deck of 52 playing cards?
Notice that the event not a spade is the complement of drawing a spade.
So, P of not spade is equal to 1 minus P of spade.
Because there are 13 spades in the deck, that probability is 13 over 52,
the whole number one turns into 52 over 52.
So, when you subtract the 13 you get 39 over 52, and that reduces to 3/4.
Example nine.
According to the Pew Research Group, 81% of teens use social media networks.
What is the probability that a randomly selected teen does not use social media networks?
Because of the word not, we know we are to use the Complement Rule.
So, the probability of not social media user is equal
to 1 minus the probability of social media user.
In the problem, it tells us that 81% of teens use social media networks.
This is a proportion, but we can also think of it as the probability
that a randomly selected teen would use a social media network.
So, we use the decimal form of the number .81 as this probability.
Then computing 1 minus .81 gives is the answer, 0.19.
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Objective five: Multiplication Rule for Independent Events.
Definition: "Two events E and F are independent if the occurrence
of event E does not affect the probability of event F. Two events are dependent
if the occurrence of event E does affect the probability of event F."
The Multiplication Rule for Independent Events says, "If E and F are independent events,
then the probability of event E and event F occurring is given by P of E
and F equals P of E times P of F. Example 10.
A die is rolled and a coin is flipped, what is the probability that the result
of the die is a five, and the coin comes up heads?
Here note that the outcome of the die does not effect in any way the result of the coin.
Therefore, these events are independent.
The probability of five and heads is equal to the probability
of five times the probability of heads.
When a die is rolled, there are six equally likely outcomes, and one of those is a five.
Therefore, the probability of a five is 1/6.
With a coin, there are two equally likely outcomes, heads and tails.
So, the probability of getting heads is 1/2.
We multiply to obtain 1/12.
Example 11.
According to the Pew Research Group, 95% of teens use the internet.
Suppose four teens are randomly selected,
what is the possibility that all four use the internet?
Here we will use the proportion 95% as the probability
that a randomly selected teen uses the internet.
And we will use the decimal form .95 to represent this probability.
Now the event that all four use the internet means that the first teen uses the internet,
and the second uses the internet, and the third and the fourth.
So, this is an AND probability.
Furthermore, the four events are independent events,
because whether one teen uses the internet
or not does not affect whether the others use the internet.
So, we can use the Multiplication Rule for Independent Events.
Thus, the probability that all four use the internet equals the probability
that the first uses the internet times the probability that the second uses the internet,
times the probability that the third uses the internet, times the probability
that the fourth uses the internet.
Each one of these four probabilities is 0.95.
We multiply them together, that can be written as 0.95
to the fourth power, and that works out to 0.8145.
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Objective six: Conditional Probability.
Definition: The Conditional Probability denoted as P of E given F is the probability
that event E occurs, given that event F has already occurred.
There are two ways to find Conditional Probability, if E and F are any events,
then the Conditional Probability P of E given F can be found by, P of E given F equals P of E
and F divided by P of F. Or, P of E given F equals the number of outcomes in E and F divided
by the number of outcomes in F. Example 12.
According to the U.S. National Center for health Statistics, in 1997 0.2% of deaths
in the U.S. were of 25 to 34 year olds whose cause of death was cancer.
In addition, 1.97% of all people who died were 25 to 34 years old.
What is the probability that a randomly selected death is the result of cancer
if the individual is known to have been 25 to 34 years old.
So, we are finding P of death due to cancer, given 25 to 34 years old.
So, death due to cancer is playing the role of event E, and event F is 25 to 34 years old.
In the problem, it is giving us information about probabilities
and not about number of outcomes.
Therefore, we will use the formula on the left to compute our answer.
So, we need to find the probability of cancer and 25 to 34 years old,
divided by probability of 25 to 34 years old.
The 0.2% number in the problem represents the proportion of deaths that were both of 25
to 34 year old, and caused by cancer.
So, this number written as a decimal will give us the probability
for the numerator, and that's .002.
The 1.97 percent in the problem is only people who were 25 to 34 years old.
So, this number gives us the probability for the denominator.
And we write that as a decimal as 0.0197.
Dividing these two numbers gives the answer 0.1015.
Example 13.
Here again we use the data in the table showing the number
of classes a student is taking and their gender.
What is the probability that a randomly selected student is taking three classes given
that the student is male?
The key word here is given and that is what indicates
that we are to use Conditional Probability.
Also notice that the information in the problem doesn't tell us anything about probabilities,
but it does give us the number of outcomes in various events.
Therefore, we can use the second method for finding Conditional Probability.
And we say, P of three classes given male is equal to the number of people
who are taking three classes and who are male, divided by the number of people who are male.
Both of these numbers we can find in the table.
The number of people who are taking three classes and who are male is 38,
and the number of people who are male is just the total of males which is 115.
Notice that we do not use the total 247 in this problem,
because when doing conditional probability, the denominator is limited to the event
that is given, which in this case is the 115 males.
That fraction works out to 0.3304.
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Objective seven: General Multiplication Rule for Dependent Events.
If we have two events E and F, and event F does not affect event E,
that is E and F are dependent events, then the probability that E and F both occur is given
by P of E and F equals P of E times P of F given E. Example 14.
suppose two cards are randomly selected from a standard deck of 52 playing cards.
What is the probability that the first card is a club, and the second card is also a club
if the draw is done without replacement.
Without replacement means that after the first card is drawn it is set aside,
and the second card is drawn from the remaining group of cards.
The two events mentioned are dependent, because the probability
that the second card is a club is affected by whether the first card is a club.
If you draw a club first, then it makes it a little bit less likely you'll draw another club.
If you don't draw a club with the first card, then it's a little bit more likely
that you would draw a club on the second try.
So, we will use the General Multiplication Rule to say that the probability of first club
and second club equals probability of first club times probability
of second club given first club.
The probability of first club is 13 over 52,
because there are 13 clubs in the deck out of the 52 cards.
Now that first card is set aside, and so, there are only 51 cards remaining.
That's the denominator for the next fraction is 51, and the numerator is 12
because there are only 12 clubs remaining in the deck.
When we multiply these fractions it's 156 over 2,652, which reduces to 1 over 17
or you could write the answer as a decimal approximation 0.0588.
Now pause the video and try these problems.