Theory Questions – Probability

Question 1

The probabilities that two athletes, P and Q, will win a gold medal in a competition are 0.75 and 0.60, respectively. What is the probability that in the competition:

(a) both P and Q will win gold medals.

Solution

• Pr(P wins) = 0.75
• Pr(P not win) = 1 – 0.75 = 0.25
• Pr(Q wins) = 0.6
• Pr(Q will not win) = 1 – 0.6 = 0.4

Pr(Q and P win) = \(\scriptsize 0.75 \: \times \: 0.6 \\ = \scriptsize 0.45\)

(b) neither of them will win a gold medal.

Solution

Pr(neither will win) = \(\scriptsize 0.25 \: \times \: 0.4 \\ = \scriptsize 0.1\)

(c) at least one of them will win a gold medal.

Solution

⇒ \(\scriptsize Pr \left( PQ^{\prime} + QP^{\prime} + PQ\right) \\ \scriptsize = \left(0.75 \times 0.4\right) + \left(0.6 \times 0.4\right)  + \left(0.75 \times 0.6\right)\)

= 0.3 + 0.15 + 0.45

= 0.9

or

Pr(at leats one wins) = 1 – P(neither wins)

= 1 – 0.1

= 0.9