Question

Paul, Joe and Ken are playing soccer. The probability that Paul scores a goal is ¼, that of Joe scoring is 3/5 and that ken scoring a goal is 4/7. Find the probability that in a soccer game:
i. Only two scores a goal
ii. Two of them score a goal
iii. None of them score a goal
iv. At least one of them scores a goal

Answer 1

Answer on Question #56512 – Math – Statistics and Probability

Question

Paul, Joe and Ken are playing soccer. The probability that Paul scores a goal is $\frac{1}{4}$, that of Joe scoring is 3/5 and that Ken scoring a goal is 4/7. Find the probability that in a soccer game:

i. Only two scores a goal

ii. Two of them score a goal

iii. None of them score a goal

iv. At least one of them scores a goal

Solution

Let

$A =$ "Paul scores a goal", $\bar{A} =$ "Paul does not score a goal",

$B =$ "Joe scores a goal", $\bar{B} =$ "Joe does not score a goal",

$C =$ "Ken scores a goal", $\bar{C} =$ "Ken does not score a goal".

It is given that $P(A) = \frac{1}{4}$, $P(B) = \frac{3}{5}$, $P(C) = \frac{4}{7}$, hence

$$P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{4} = \frac{3}{4}, \quad P(\bar{B}) = 1 - P(B) = 1 - \frac{3}{5} = \frac{2}{5}, \quad P(\bar{C}) = 1 - P(C) = 1 - \frac{4}{7} = \frac{3}{7}.$$

We can use product rule to fill the next table.

i. The probability that only two score ball equals

$$P(AB\bar{C}) + P(\bar{A}BC) + P(A\bar{B}C) = \frac{9}{140} + \frac{36}{140} + \frac{8}{140} \approx 0.3786.$$

ii. The probability that two of them score ball equals

$$P(AB\bar{C}) + P(\bar{A}BC) + P(A\bar{B}C) + P(ABC) = \frac{9}{140} + \frac{36}{140} + \frac{8}{140} + \frac{12}{140} \approx 0.4643.$$

iii. The probability that none of them score ball equals

$$P(\bar{A}\bar{B}\bar{C}) = \frac{18}{140} \approx 0.1286.$$

iv. The probability that at least one of them scores ball equals

$$\begin{array}{l} P(ABC) + P(AB\bar{C}) + P(A\bar{B}C) + P(A\bar{B}\bar{C}) + P(\bar{A}BC) + P(\bar{A}B\bar{C}) + P(\bar{A}\bar{B}C) = \\ = 1 - P(\bar{A}\bar{B}\bar{C}) = 1 - \frac{18}{140} \approx 0.8714. \end{array}$$

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