Question

Suppose you have four extra tickets to a football game and you are trying to determine which of your five closest friends to take with you: Mark, Samuel, Elizabeth, James, and Mary. You decided to randomly choose four of them to attend the game with you. What is the probability that you will choose at least one friend whose name starts with the letter M? 0 2/5 4/5 1 2

Answer 1

Solution

Back-up Theory

Probability of an event E, denoted by P(E) = n/N ……………………………………....................................……........……(1)

where

n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and

N = n(S) = Total number all possible outcomes/cases/possibilities.

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}…...............................................…(2)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN…...............................................…. (2a)

Number of ways of selecting r things out of n things when p specific things must be selected is given by ^{n - p}C_{r - p}....... (3a)

Number of ways of selecting r things out of n things when p specific things must NOT be selected is given by ^{n - p}C_r....(3b)

Number of ways of selecting r things out of n things when p specific things must be selected and q specific things must NOT be selected is given by ^{n – p - q}C_{r - p}................................................................................................................. (3c)

Now, to work out the solution,

Out of 5 names, two names, namely, Mark and Mary start with the letter M.

‘Choose at least one friend whose name starts with the letter M’ =>

Exactly one friend whose name starts with the letter M, i.e., Mark is selected but Mary is not selected

Or,

Both Mark and Mary are selected.

Vide (3c), number of selections for the first case is: ³C₃ = 1

Vide (3a), number of selections for the second case is: ³C₂ = 3.

So, there are 4 selections possible where at least one friend whose name starts with the letter M. Vide (1), n = 4

Vide (2), 4 out of 5 friends can be selected in ⁵C₄ = 5 ways and vide (1), N = 5

Hence, vide (1), probability that you will choose at least one friend whose name starts with the letter M

= 4/5 Third Optiion Answer

DONE