Review - Discrete Probability Distributions.pdf

MDM4U

Situation

Distribution

𝑷(𝒙)

𝑬(𝒙)

Approximately 5% of the first batch of engines off a new production line have flaws. Six engines are randomly selected from the production line, and tested for flaws. Identify the probability distribution of the number of flawed engines.

In a swim meet, there are 16 competitors, 5 of whom are from the Eastern Swim Club. There are 5 swimmers in the first heat of the swim meet. Identify the probability distribution of the number of swimmers from Eastern Swim Club in the first heat for the meet.

Jamaal has a success rate of 68% for scoring on free throws in basketball. Identify the probability distribution for the waiting time before he misses the basket on a free throw.

Earlier this year, 520 of the 2000 seals in a particular sea were caught and tagged. On a recent survey, 125 seals were caught, and the number of tagged seals were counted. Identify the probability

distribution of the number of tagged seals counted.

Krakked Korn cereal is having a promotion where they put one of seven different baseball cards in their cereal boxes. If the cards distributed randomly in the cereal boxes, identify the probability distribution of the baseball cards.

Suppose 60% of the families in a town own computers. Eight families are surveyed at random. Identify the probability

Situation

Distribution

𝑷(𝒙)

𝑬(𝒙)

When he was in high school, Mr. L had a job doing telephone surveys. Of the people he called, nine out of ten were not willing to do the survey. Identify the probability distribution of the number of unwilling people he had to call before doing a survey.

A baseball player has a batting average of 0.320. He usually has 3 times at bat each game. Identify the probability distribution of the number of hits he has in a game.

Suppose that an intersection you pass on your way to school has a traffic light that is green for 40 seconds and red for 60 seconds. Identify the probability

distribution of the number of days it will take for you to get a green at the traffic light on your way to school.

In a mathematics class of 20 students, 5 bilingual. A group of 5 students is selected from the class. Identify the probability distribution of the number of bilingual students in that group.

A telemarketing firm has a list of all of the phone numbers in Oakville. There are approximately 80, 000 numbers on this list. They choose numbers at random from this list to call. Identify the probability

distribution of the phone numbers.

MDM4U

Situation

Distribution

𝑷(𝒙)

𝑬(𝒙)

Approximately 5% of the first batch of engines off a new production line have flaws. Six engines are randomly selected from the production line, and tested for flaws. Identify the probability

distribution of the number of flawed engines.

Binomial

𝑝 = 0.05 𝑞 = 0.95 𝑛 = 6

𝑥 = number of flawed engines

𝑃(𝑥) =6𝐶𝑥(0.05)𝑥(0.95)6−𝑥

𝐸(𝑥) = 6(0.05) = 0.3

In a swim meet, there are 16 competitors, 5 of whom are from the Eastern Swim Club. There are 5 swimmers in the first heat of the swim meet. Identify the probability distribution of the number of swimmers from Eastern Swim Club in the first heat for the meet.

Hypergeometric

𝑎 = 5 𝑛 = 16

𝑟 = 5

𝑥 = number of swimmers from Eastern Swim Club in the first heat

𝑃(𝑥) =(5 𝐶𝑥)(11 𝐶5−𝑥) (16 𝐶5)

𝐸(𝑥) =5(5) 16 = 1.56 ≈ 2

Jamaal has a success rate of 68% for scoring on free throws in basketball. Identify the probability distribution for the waiting time before he misses the basket on a free throw.

Geometric

𝑝 = 0.32

(because he misses the last one!)

𝑞 = 0.68

𝑥 = number of shots before he misses

𝑃(𝑥) = (0.68)𝑥_(0.32

𝐸(𝑥) =0.68 0.32 = 2.125 ≈ 2

Earlier this year, 520 of the 2000 seals in a particular sea were caught and tagged. On a recent survey, 125 seals were caught, and the number of tagged seals were counted. Identify the probability

Hypergeometric

𝑎 = 520 𝑛 = 2000

𝑟 = 125

𝑥 = number of tagged seals that were caught in the 125 trials

𝑃(𝑥) =(125 𝐶𝑥)(1875 𝐶520−𝑥) (2000 𝐶520)

distribution of the number of tagged seals counted.

Krakked Korn cereal is having a promotion where they put one of seven different baseball cards in their cereal boxes. If the cards distributed randomly in the cereal boxes, identify the probability distribution of the baseball cards.

Uniform 𝑃(𝑥) =1

7

None for this case (but there may be expected

values for uniform distributions)

Suppose 60% of the families in a town own computers. Eight families are surveyed at random. Identify the probability distribution of the number of families selected who own computers.

Binomial

𝑝 = 0.6 𝑞 = 0.4 𝑛 = 8

𝑥 = number of families who own computers out of the eight surveyed

𝑃(𝑥) = (8 𝐶𝑥)(0.60)𝑥(0.40)8−𝑥

𝐸(𝑥) = 8(0.6) = 4.8 ≈ 5

When he was in high school, Mr. L had a job doing telephone surveys. Of the people he called, nine out of ten were not willing to do the survey. Identify the probability

distribution of the number of unwilling people he had to call before doing a survey.

Geometric

𝑝 = 1 10= 0.1

𝑞 = 9 10= 0.9

𝑥 = number of unwilling people called

𝑃(𝑥) = (9 10)

𝑥

(1 10)

𝐸(𝑥) =0.9 0.1 = 9

A baseball player has a batting average of 0.320. He usually has 3 times at bat each game. Identify the probability distribution of the number of hits he has in a game.

Binomial

𝑛 = 3 𝑝 = 0.320

𝑞 = 0.68

𝑥 = the number of hits out of 3

𝑃(𝑥) = (3 𝐶𝑥)(0.320)𝑥(0.68)𝑛−𝑥

Suppose that an

intersection you pass on your way to school has a traffic light that is green for 40 seconds and red for 60 seconds. Identify the probability distribution of the number of days it will take for you to get a green at the traffic light on your way to school.

Geometric

𝑝 = 0.4 𝑞 = 0.6

𝑥 = number of days before you get a green light

𝑃(𝑥) = (0.6)𝑥(0.4)

𝐸(𝑥) =0.6 0.4

=3 2 = 1.5

≈ 2

In a mathematics class of 20 students, 5 bilingual. A group of 5 students is selected from the class. Identify the probability

distribution of the number of bilingual students in that group.

Hypergeometric

𝑎 = 5 𝑛 = 20

𝑟 = 5

𝑥 = number of bilingual students in the group of 5

𝑃(𝑥) =(5 𝐶𝑥)(15 𝐶5−𝑥) (20 𝐶5)

𝐸(𝑥) =(5)(5) 20

=25 20 = 1.25

≈ 1

A telemarketing firm has a list of all of the phone numbers in Oakville. There are approximately 80, 000 numbers on this list. They choose numbers at random from this list to call. Identify the probability distribution of the phone numbers.

Uniform 𝑃(𝑥) = 1

80000

None for this case (but there may be expected

values for uniform distributions)

Mail order marketing companies have a response rate of 15% to their advertising flyers. Identify the probability distribution of the number of responses they receive to a mail-out of 100 flyers.

Binomial

𝑝 = 0.15 𝑞 = 0.85 𝑛 = 100

𝑥 = number of responses they receive

𝑃(𝑥) = (100 𝐶𝑥)(0.15)𝑥(0.85)100−𝑥