Chapter 5:

Page 198:

Question 7: A die is rolled until the first time T that a six turns up.

  1. What is the probability distribution for T?
  2. Find P(T > 3).
  3. Find P(T > 6|T > 3).

Solution:

  1. What is the probability distribution for T?

Let P(6) be the probability of rolling a six on any single roll.

p = 1/6

Let P(Not6) be the probability of not rolling a six on any single roll.

P(Not6) = 5/6

Probability that a six appears for the first time on the T-th roll is:

\[P(T=t) = (P!=6)^{t-1} * P(6)\]

The probability distribution for T is:

\[P(T=t) = (5/6)^{t-1} * ((1/6)\]

This is a is a geometric distribution where t is the number of rolls until the first six appears.

  1. Find P(T > 3).
P_roll <- 1/6

n <- (3 - 1)

P_3 <- pgeom(n, P_roll, lower.tail = F)

P_3
## [1] 0.5787037
  1. Find P(T > 6|T > 3).
P_roll <- 1/6

n <- (6 - 1)

P_6 <- pgeom(n, P_roll, lower.tail = F)

P_final <- P_3 * P_6/P_3

P_final
## [1] 0.334898

=============================================================== ### Discrete Probability Distribution: ### Page # 36 ### Question # 8

A student must choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology. What are the respective probabilities that she chooses art, geology, and psychology?

Solution:

To choose one of the subjects, art, geology, or psychology, as an elective. She is equally likely to choose art or psychology and twice as likely to choose geology.

Let set the probability as

P(A) be the probability of choosing art.

P(G) be the probability of choosing geology.

P(P) be the probability of choosing psychology.

Since the student is equally likely to choose art or psychology, then

P(A)=P(P)=x

Also the student is twice as likely to choose geology, then

P(G)=2x

The probabilities of all possible outcomes must sum up to 1,

P(A)+P(G)+P(P)=1

substituting the values below

x + 2x + x = 1 4x = 1 x = 1/4

Now find the probabilities of each subject

P(Art)=P(Psychology)

P(A) = P(P) = 1/4

P(G) = 2 * 1/4 = 1/2

P(Geology) = 1/2