Two players take one penalty kick each. The probability that the first player scores is \(\frac{2}{3}\) and the probability that the second player scores is \(\frac{3}{5}\). The two kicks are independent of each other. What is the probability that exactly one of the two players scores?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\frac{7}{15}\)
“Exactly one scores” happens in two separate ways: the first scores and the second misses, or the first misses and the second scores. For independent kicks each way is a product, and the two ways are added.
P(first misses) \(\displaystyle =1-\frac{2}{3}=\frac{1}{3}\); P(second misses) \(\displaystyle =1-\frac{3}{5}=\frac{2}{5}\)
First scores, second misses \(\displaystyle =\frac{2}{3}\times\frac{2}{5}=\frac{4}{15}\)
First misses, second scores \(\displaystyle =\frac{1}{3}\times\frac{3}{5}=\frac{3}{15}\)
P(exactly one scores) \(\displaystyle =\frac{4}{15}+\frac{3}{15}=\frac{7}{15}\)