Probability calculations

Question 1

A standard six-sided die is rolled.

(a) What is the probability of rolling an even number?

(b) What is the probability of rolling a number greater than 4?

(c) What is the probability of not rolling a 3?

Answer 1

Solution: Sample space:
𝑆 = {1, 2, 3, 4, 5, 6}, so 𝑛(𝑆) = 6.

(a) Even numbers: {2, 4, 6}, so 𝑛(𝐴) = 3.

𝑃(𝐴) = 𝑛(𝐴) / 𝑛(𝑆) = 3 / 6 = 1/2.

(b) Numbers greater than 4: {5, 6}, so 𝑛(𝐵) = 2.

𝑃(𝐵) = 𝑛(𝐵) / 𝑛(𝑆) = 2 / 6 = 1/3.

(c) Not rolling a 3: Complement Rule:
𝑃(𝐴′) = 1 − 𝑃(rolling 3) = 1 − 1/6 = 5/6.

Question 2

Question: If two fair coins are flipped:
(a) Write out the sample space.
(b) What is the probability of getting at least one heads?

Answer 2

Solution:

(a) Sample space:
𝑆 = {HH, HT, TH, TT}, so 𝑛(𝑆) = 4.

(b) At least one heads: 𝐴 = {HH, HT, TH}, so 𝑛(𝐴) = 3.

𝑃(𝐴) = 3 / 4 = 0.75.

Question 3

If an event A has a probability of 0.65, what is the probability of A’ (not A)?

Answer 3

Solution: Using the complement rule:

𝑃(𝐴′) = 1 − 𝑃(𝐴)
𝑃(𝐴′) = 1 − 0.65 = 0.35.

Question 4

Question: Given the probability distribution:

X 0 1 2 3
P(X) 2k k 3k 4k
(a) Find the value of 𝑘 so that this is a valid probability distribution.
(b) Find 𝑃(𝑋 < 2).
(c) Find 𝑃(𝑋 ≥ 1).

Answer 4

Solution:

(a) Since total probability must equal 1:
2𝑘 + 𝑘 + 3𝑘 + 4𝑘 = 1
10𝑘 = 1
𝑘 = 1/10 = 0.1.

(b) Find 𝑃(𝑋 < 2) (i.e., 𝑃(0) + 𝑃(1)):
𝑃(0) = 2𝑘 = 2(0.1) = 0.2
𝑃(1) = 𝑘 = 0.1
𝑃(𝑋 < 2) = 0.2 + 0.1 = 0.3.

(c) Find 𝑃(𝑋 ≥ 1) (i.e., 𝑃(1) + 𝑃(2) + 𝑃(3)):
𝑃(2) = 3𝑘 = 3(0.1) = 0.3
𝑃(3) = 4𝑘 = 4(0.1) = 0.4
𝑃(𝑋 ≥ 1) = 0.1 + 0.3 + 0.4 = 0.8.

Question 5

Suppose P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2.
Find 𝑃(𝐴 ∪ 𝐵).

Answer 5

Solution: Using the Addition Rule for Non-Mutually Exclusive Events:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴 ∪ 𝐵) = 0.4 + 0.5 − 0.2 = 0.7.

Question 6

Question: If P(C) = 0.3 and P(D) = 0.6, and events C and D cannot happen at the same time, find 𝑃(𝐶 ∪ 𝐷).

Answer 6

Solution: For mutually exclusive events:
𝑃(𝐶 ∪ 𝐷) = 𝑃(𝐶) + 𝑃(𝐷)
𝑃(𝐶 ∪ 𝐷) = 0.3 + 0.6 = 0.9.

Question 7

Question: If P(A) = 0.6 and P(B) = 0.3, and A and B are independent, what is 𝑃(𝐴 ∩ 𝐵)?

Answer 7

Solution: Using the Multiplication Rule for Independent Events:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵) = 0.6 × 0.3 = 0.18.

Question 8

Question: If P(A | B) = 0.25 and P(B) = 0.2, find 𝑃(𝐴 ∩ 𝐵).

Answer 8

Solution: Using the Multiplication Rule for Dependent Events:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 | 𝐵) × 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵) = 0.25 × 0.2 = 0.05.

Question 9

A company has 300 employees, each selecting one of the following performance bonuses:

84 employees chose cash
97 employees chose shares
119 employees chose profit-sharing

The company also has employees working in different departments:

56 of the employees who chose cash work in production
The company has 140 employees in administration, 68 of whom chose shares

Find the following probabilities:

(a) The probability of an employee choosing a cash bonus.
(b) The probability of an employee choosing a tax-free bonus (shares).
(c) The probability of an employee being in production and choosing a cash bonus.
(d) The probability of an employee choosing shares given that they work in administration.
(e) The probability of a production worker choosing cash given that they already chose cash.

Answer 9

(a) 𝑃(𝐶𝑎𝑠𝑕) = 84 / 300 = 0.28
(b) 𝑃(𝑆ℎ𝑎𝑟𝑒𝑠) = 97 / 300 = 0.3233
(c) 𝑃(𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 ∩ 𝐶𝑎𝑠𝑕) = 56 / 300
(d) 𝑃(𝑆ℎ𝑎𝑟𝑒𝑠 | 𝐴𝑑𝑚𝑖𝑛) = 68 / 140 = 0.4857
(e) 𝑃(𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 | 𝐶𝑎𝑠𝑕) = 56 / 84 = 0.6667