Question

In 2019, approximately 97.4% of all the runners who started the Boston Marathon (in Boston, Massachusetts, USA) were able to complete the 42.2 km (26.2 mile) race. If 100 runners are chosen at random, find the probability that exactly 97 of them finished the marathon​

Answer 1

Answer:

The probability that exactly 97 of them finished the marathon​ is 0.221.

Step-by-step explanation:

We are given that in 2019, approximately 97.4% of all the runners who started the Boston Marathon were able to complete the 42.2 km race.

100 runners are chosen at random.

The above situation can be represented through the binomial distribution;

$P(X = x) = \binom{n}{r} \times p^{r} \times (1-p)^{n-r};x = 0,1,2,3,......$

where, n = number of trials(or samples) taken = 100 runners

           r = number of success = exactly 97

           p = probability of success which in our question is the probability

                 that runners finished the marathon​, i.e; p = 0.974

So, X ~ Binom(n = 100, p = 0.974)

Now, the probability that exactly 97 of them finished the marathon​ is given by = P(X = 97)

         P(X = 97) = $\binom{100}{97} \times 0.974^{97} \times (1-0.974)^{100-97}$

                          = $161700 \times 0.974^{97} \times 0.026^{3}$

                          = 0.221

Hence, the required probability is 0.221.