9-7 Probability of Multiple Events

1. 9-7Probability of Multiple Events

2. Review-- A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles and 12 red marbles. Suppose you pick one marble at random. Find each probability. 7/36 P(yellow) = 25/36 P(not blue) = P(green or red) = 1/2

3. When the outcome of one event affects the outcome of a second event, the two events are dependent events. • Example– Select a marble from a bag that contains marbles of two colors. Put the marble aside, and select a second marble from the bag. • When the outcome of one event does not affect the outcome of the second event, the two events are independent events.

4. Examples of Independent events-- • Examples— • Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die. • Choosing a marble from a jar AND landing on heads after tossing a coin. • Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card. • Rolling a 4 on a single 6-sided die, AND then rolling a 1 on second roll of the die. • To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, then multiply the probabilities. P(A and B) = P(A) * P(B)

5. A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times? P(red and red) = P(red) * P(red) = 1/5 * 1/5 = 1/25

6. A coin is tossed and a single 6 sided die is rolled. Rind the probability of landing on the head side of the coin and rolling a 3 on the die. • P(head and 3) = P(head) * P(3) = 1/2* 1/6 = 1/12

7. Mutually Exclusive Events • Two events are non-mutually exclusive if they have one or more outcomes in common. • Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common). P(A) P(A) P(B) P(B) In the Venn Diagram above, the probabilities of events A and B are represented by two disjoint sets (i.e., they have no elements in common). In the Venn Diagram above, the probabilities of events A and B are represented by two intersecting sets (i.e., they have some elements in common).

8. Example-- • A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a 5 or a king? • Possibilities— • The card chosen can be a 5. • The card chosen can be a king. • Since they are mutually exclusive— • P(5 or king) = P(5) + P(king)= 4/52 + 4/52 = 8/52

9. Example-- • A single 6-sided die is rolled. What is the probability of rolling an odd number or an even number? • Possibilities— • The number rolled can be an odd number. • The number rolled can be an even number. • Since they are mutually exclusive— • P(odd or even) = P(odd) + P(even)= 1/2 + 1/2 = 1

10. Example-- • A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a club or a king? • Possibilities— • The card chosen can be a club. • The card chosen can be a king. • The card chosen can be a club and a king. • Since they are NOT mutually exclusive— • P(Club or King) = P(Club) + P(king) – P(club and King) • = 13/52 + 4/52 – 52/2704 • = 13/52 + 4/52 – 1/52 • = 16/52 = 4/13

11. Example-- • A single 6 sided die is rolled. What is the probability of rolling a 5 or an odd number. • Possibilities— • The number rolled can be a 5. • The number rolled can be an odd number. • The number rolled can be a 5 and an odd number. • Since they are NOT mutually exclusive— • P(5 or odd) = P(5) + P(odd) – P(5 and odd) • = 1/6 + 3/6 – 3/6 • = 1/6