Math I

 Probability is the chance that something will happen.  Probability is most often expressed as a fraction, a decimal, a percent, or can also be written out in words.  To determine the probability… P(event) = number of true outcomes total number of equally likely outcomes

 Independent : Two events are independent if the occurrence of one has no effect on the occurrence of the other…

 Example 1: (probability of two events) What is the probability of drawing a king and then an ace from a standard 52 card deck with replacement? P(King, Ace) =  Example 2 : What is the probability of flipping heads on a coin three times in a row? P(H, H, H) =

 Example 3: A die is rolled twice. What’s the probability of rolling a 2 and then an even number? Solution:  Example 4: You spin the spinner 3 times. What is the probability of spinning a 4, a 3 and then a 1? Solution:

 Dependent: Two events such that the occurrence of one affects the occurrence of the other. P(A and B) = P(A) P(B|A) **P(B|A) = means the probability of B given that event A has already occurred.

 Example 1 : What is the probability of drawing a King and then an Ace without replacement ? P(King, Ace) =

 Example 2: You randomly select two marbles from a bag that contains 14 green, 7 blue, and 9 red marbles. What is the probability that the first marble is blue and the second marble is not blue if you do not replace the first marble? Solution:

 Example 3: Your teacher passes around a basket with 6 red erasers, 9 blue erasers, and 7 green erasers. If you and your two neighbors are the first to randomly select an eraser, what is the probability that all three of you select green erasers? Solution : P(A) and P(B|A) and P(C|A and B)

 The table shows the number of males and females with certain hair colors. Find …  A) the probability that a listed person has red hair  B) the probability that a female has red hair Brown hair Blonde hair Red hair Black hair OtherMale Female

 P(red hair) = # of people with red hair total # of people  P(red hair | female) = # of red hair females total # of females

 When you consider the outcomes for either of two events A and B, you form the union of A and B.

 When you consider only the outcomes shared by both A and B, you form the intersection of A and B.

 When the sets of A and B have nothing in common (no intersection) then they are considered mutually exclusive events.

 If A and B are mutually exclusive events (one event does not have anything in common with the other), then… P(A or B) = P(A) + P(B)

 A die is rolled one time. What is the probability of rolling a 2 or a 6? Solution: P(A or B) = P(A) + P(B) =

 A card is randomly selected out of a standard deck of 52 cards. What is the probability that it is a 2 or a king? P(A or B) = P(A) + P(B) =

 If A and B are not mutually exclusive, then there are some outcomes in common.  Therefore, the intersection of A and B are counted twice when P(A) and P(B) are added.  So, P(A and B) must be subtracted once from the sum… P(A or B) = P(A) + P(B) – P(A and B)

 A die is rolled one time. What is the probability of rolling an odd number or a prime number?  Odd = 1, 3, 5 P(A) = Prime = 2, 3, 5 P(B) =  Odd and prime = 3, 5 P(A and B) =

 A card is randomly selected from standard deck of 52 cards. What is the probability that it is a red card or a king?  Red cards = 26 = Kings = 4 =  Red Kings = 2 =

 The probability that it will rain today is 40%. The probability that is will rain tomorrow is 30%. The probability that it will rain both days is 20%. Find the probability that it will rain either today OR tomorrow.  Solution: P(A ) + P(B) – P(A and B) P(today) + P(tomorrow) – P(today and tomorrow) =

 The expected value is often referred to as the “long-term” average or mean. expected value  This means that over the long term of doing an experiment over and over, you would expect this average.  To find the expected value or long term average, simply multiply each value of the random variable by its probability and add the products.

 Suppose that the following game is played. A man rolls a die. If he rolls a 1, 3, or 5, he loses $3, if he rolls a 4 or 6, he loses $2, and if he rolls a 2, he wins $12. What gains or losses should he expect on average? (What is his expected value ?)

 We must find the probabilities of each outcome. We can make a chart to help us see this. Possible Losses/Gains Probability P(losses/gains) -$3 -$2 $12

 Now, we multiply the probability for each outcome by the amount of money either gained or lost for that outcome.  Expected value of rolling a 1, 3, or 5: ________________  Expected value of rolling a 4 or 6: ________________  Expected value of rolling a 2: _______________

 To find the expected value for the entire game (the answer), simply add up the expected value for each outcome.  Expected Value =  This means that the man playing this game is expected to lose an average of $0.17 each game he plays.