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Probability & Expected Value

Probability & Expected Value . Math I. MM1D2: Students will use the basic laws of probability. Simple Probability …(review). 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.

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Probability & Expected Value

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  1. Probability & Expected Value Math I

  2. MM1D2:Students will use the basic laws of probability

  3. Simple Probability…(review) • 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

  4. Independent Events • Independent: Two events are independent if the occurrence of one has no effect on the occurrence of the other…

  5. Examples: • 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) =

  6. More Examples… • 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:

  7. MMD12b:Find the probabilities of dependent events

  8. Dependent Events: • 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.

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

  10. More Examples… • 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:

  11. 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)

  12. Probability from a Table: • 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

  13. Solution: • P(red hair) = # of people with red hair total # of people • P(red hair | female) = # of red hair females total # of females

  14. Probability of Compound Events

  15. Compound Events: Union • When you consider the outcomes for either of two events A and B, you form the union of A and B.

  16. Compound Events: Intersection • When you consider only the outcomes shared by both A and B, you form the intersection of A and B.

  17. Mutually Exclusive Events: • When the sets of A and B have nothing in common (no intersection) then they are considered mutually exclusive events.

  18. Probability of Mutually Exclusive Compound 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)

  19. Example 1: • 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) =

  20. Example 2: • 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) =

  21. Probability of Compound Events NOT Mutually Exclusive: • 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)

  22. Example 3: • 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) =

  23. Example 4: • 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 =

  24. Extension: • 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) =

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