Question

Suppose you and a friend each choose at random an integer between 1 and 8, inclusive. For example, some possibilities are (3,7), (7,3), (4,4), (8,1), where your number is written first and your friend’s number second. Find the following probabilities.
a. p(you pick 5 and your friend picks 8)
b. p(sum of the two numbers picked is < 4)
c. p(both numbers match)

Answer 1

Answer and explanation:

Given : Suppose you and a friend each choose at random an integer between 1 and 8, inclusive.

The sample space is

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)  (1,7) (1,8)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)   (2,7) (2,8)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)   (3,7) (3,8)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)   (4,7) (4,8)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)   (5,7) (5,8)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)   (6,7) (6,8)

(7,1) (7,2) (7,3) (7,4) (7,5) (7,6)   (7,7) (7,8)

(8,1) (8,2) (8,3) (8,4) (8,5) (8,6)   (8,7) (8,8)

Total number of outcome = 64

To find : The following probabilities ?

Solution :

The probability is given by,

$\text{Probability}=\frac{\text{Favorable outcome }}{\text{Total outcome}}$

a) p(you pick 5 and your friend picks 8)

The favorable outcome is (5,8)= 1

$\text{Probability}=\frac{1}{64}$

b) p(sum of the two numbers picked is < 4)

The favorable outcome is (1,1), (1,2), (2,1)= 3

$\text{Probability}=\frac{3}{64}$

c) p(both numbers match)

The favorable outcome is (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8) = 8

$\text{Probability}=\frac{8}{64}$

$\text{Probability}=\frac{1}{8}$