6.4 Find Probabilities of Compound Events

Vocabulary Compound event: Combines 2+ events Using words and or the word or Mutually exclusive events: no common outcomes Overlapping events: 1+ common outcome

Vocabulary Continued Independent events: 1 event has no effect on the other Color of your teacher’s hair and the grade on a test Dependent events: 1 event affects the other event Bad weather and number of automobile accidents Conditional Probability: the probability that one event will occur because another one occurred

Let’s Review If 1 even occurs m ways & another n ways, then the # of both occurring is m*n. Multiplication Counting Principle Addition Counting Principle If groups formed with nothing in common, total # of possibilities is sum of #’s. Permutations Formula nPr = Combinations Formula nCr =

Probability of mutually exclusive events If A and B are mutually exclusive, then P(A or B)= P(A) + P(B). Probability of overlapping events If A and B are overlapping, then P(A or B) = P(A) + P(B) – P(A and B) If A and B are independent, then P(A and B)= P(A) * P(B). Probability of Independent events Probability of Dependent events If A and B are dependent, then P(A and B)= P(A) * P(B given A).

You roll a number cube. Find the probability that you roll a 2 or an odd number. SOLUTION Because 2 is an even number, rolling a 2 and rolling an odd number are mutually exclusive events. P(2 or odd) = P(2) + P(odd) 6 1 = + 3 4 = 6 = 2 3

Find the probability of A or B You roll a number cube. Find the probability that you roll an even number or a prime number. SOLUTION Because 2 is both an even number and a prime number, rolling an even number and rolling a prime number are overlapping events. There are 3 even numbers, 3 prime numbers, and 1 number that is both. P(even or prime) = P(even) + P(prime) – P(even and prime) 3 6 = + – 1 5 6 =

1. You roll a number cube. Find the probability that 1. You roll a number cube. Find the probability that you roll a 2 or a 5. 1 3 ANSWER 2. You roll a number cube. Find the probability that you roll a number less than 4 or an odd number. 2 3 ANSWER

Find the probability of A and B BUS SCHEDULE You take a city bus from your neighborhood to a location within walking distance of your school. The express bus arrives at your neighborhood between 7:30 and 7:36. The local bus arrives at your neighborhood between 7:30 and 7:40. You arrive at the bus stop at 7:33. Find the probability that you have missed both the express bus and the local bus. SOLUTION The events are independent. The arrival of one bus does not affect the arrival of the other bus.

There are 6 minutes when the express bus can arrive There are 6 minutes when the express bus can arrive. You are not at the bus stop for 3 of those minutes. There are 10 minutes when the local bus can arrive. You are not at the bus stop for 3 of those minutes. P(you miss express bus) P(you miss local bus) 3 6 = 1 2 = = 3 10

Find the probability of A and B Multiply the probabilities of the two events: 3 10 1 2 = 3 20 = P(you miss both buses) ANSWER The probability that you miss the express bus and the local bus is 3 20

Find the probability of A and B PEN COLORS A box contains 3 blue pens and 5 black pens. You choose one pen at random, do not replace it, then choose a second pen at random. What is the probability that both pens are blue? SOLUTION Because you do not replace the first pen, the events are dependent. Before you choose a pen, there are 8 pens, and 3 of them are blue. After you choose a blue pen, there are 7 pens left and 2 of them are blue. P(blue and then blue) = P(blue) P(blue given blue) 3 8 2 7 = = 6 56 = 3 28

3. MARBLES A bag contains 4 red, 5 green, and 2 blue marbles 3. MARBLES A bag contains 4 red, 5 green, and 2 blue marbles. You randomly draw 2 marbles, one at a time. Find the probability that both are red if: a. You replace the first marble. ANSWER 16 121 b. You do not replace the first marble. ANSWER 6 55