Question

In the first five hockey games of the season, Pintsize Penguins had the following numbers of goals scored against them: 4, 6, 4, 8, 3.

Suppose you select two games at random from the first five games.

i. Calculate the probability that the number of goals conceded was 5 or less for the first game selected. Express your answer as a fraction in simplest form.

ii. Calculate the probability that the number of goals conceded was 5 or less for the second game selected, given that the number of goals conceded was 5 or less for the first game selected. Express your answer as a fraction in simplest form.

iii. Calculate the probability that the number of goals conceded was 5 or less for both selected games. Express your answer as a fraction in simplest form.

Answer 1

There are 3 games with the number of goals less than 5 (4, 4, 3)

There are 2 games with the number of goals more than 5 (6, 8)

Answer i)

No. of ways in the first game selected has goals less than 5 = ₃C₁

Corresponding to it, number of ways in which second game can be selected = ₄C₁

So, number of ways in which 2 games can be selected with first game having goals less than 5 = ₃C₁*₄C₁ = 12

Total number of ways of selecting 2 games out of 5 = ₅P₂ = 20

P(G1 less than 5) = 12/20

P(G1 less than 5) = 3/5

Answer ii)

P(G2 less than 5 | G1 less than 5) = P(G2 less than 5 AND G1 less than 5)/P(G1 less than 5)

No. of ways in the first game selected has goals less than 5 = ₃C₁

Corresponding to it, number of ways in which second game selected has goals less than 5 = ₂C₁

Number of ways in which 2 games can be selected with both games having goals less than 5 = ₃C₁*₂C₁ = 6

Total number of ways of selecting 2 games out of 5 = ₅P₂ = 20

Thus, P(G2 less than 5 AND G1 less than 5) = 6/20

P(G2 less than 5 AND G1 less than 5) = 3/10

P(G2 less than 5 | G1 less than 5) = (3/10)/(3/5)

P(G2 less than 5 | G1 less than 5) = 1/2

Answer iii)

P(G2 less than 5 AND G1 less than 5) = 3/10 (Refer Answer ii)