Question

Mr. Bobby Fischer (1943-2008) is one of the finest Chess players ever known. Just at the age of 18, he was supposed to play 8 different championships in a year. To become the World champion, he must win 6 of them. What is the probability that he won at least 5 of them?

A. 0.052
B. 0.189
C. 0.711
D. 0.945

Answer 1

The probability that Mr. Bobby Fischer won at least 5 out of the 8 championships is 0.945, corresponding to option D.

Step-by-step explanation:

To find the probability, we can use the binomial probability formula. In this case, Mr. Fischer must win at least 5 out of 8 championships. The probability of winning exactly k out of n championships is given by the formula:

`\[ P(X = k) = \binom{n}{k} * p^k * (1-p)^(n-k) \]`

where
`\( \binom{n}{k} \)` is the binomial coefficient, p is the probability of success in a single trial, and n is the number of trials.

In this scenario, n = 8 championships, and Mr. Fischer must win 5, 6, 7, or 8 of them. So, we calculate the probabilities for each case and sum them up:

P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

This probability is then compared with the given options. In this case, the calculated probability matches option D, which is 0.945.

Understanding the binomial probability formula and its application in counting successes in a fixed number of independent trials is crucial for solving such probability problems.