Question

An individual claims to have extrasensory perception (ESP). As a test, a fair coin is flipped ten times, and he is asked to predict in advance the outcome. Our individual gets seven out of ten correct. What is the probability he would have done at least this well if he had no ESP? (Explain why the relevant probability is \(P[X \geq 7\\}\) and not \(P\\{X=7\\} .)\)

Short answer

The probability of the individual getting at least 7 out of 10 correct predictions by chance alone, without having extrasensory perception (ESP), is \(P(X \geq 7)\). To find this cumulative probability, we calculated the binomial probabilities for each possible number of successes (7, 8, 9, and 10) and summed them: \[P(X \geq 7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)\] Using the binomial probability formula, we found the final probability \(P(X \geq 7)\).

Step-by-step solution

Define variables and probabilities

Let X be the number of correct predictions by the individual. X follows a binomial distribution with parameters n (number of trials) and p (probability of success in a single trial). Here, n=10 (coin is flipped 10 times) and p=0.5 (since it's a fair coin, there's a 50% chance of guessing each side correctly).

Use the binomial probability formula

The binomial probability formula is given by: \[P(X=k) = {{n}\choose{k}} p^k (1-p)^{n-k}\] where \(P(X=k)\) is the probability of having k successes (in our case, correct predictions), \({{n}\choose{k}}\) is the number of ways to choose k successes from n trials, and p is the probability of success in each trial.

Calculate the probabilities for each possible number of successes (7, 8, 9, and 10)

Using the binomial probability formula, we will calculate the probabilities for each possible number of correct predictions (7, 8, 9, and 10): - For k=7: \(P(X=7) = {{10}\choose{7}} 0.5^7 (1-0.5)^{10-7}\) - For k=8: \(P(X=8) = {{10}\choose{8}} 0.5^8 (1-0.5)^{10-8}\) - For k=9: \(P(X=9) = {{10}\choose{9}} 0.5^9 (1-0.5)^{10-9}\) - For k=10: \(P(X=10) = {{10}\choose{10}} 0.5^{10} (1-0.5)^{10-10}\)

Sum the probabilities to find the cumulative probability

To find the probability of at least 7 correct predictions (P[X ≥ 7]), we must sum the probabilities calculated in Step 3: \[P(X \geq 7) = P(X=7) + P(X=8) + P(X=9) + P(X=10)\]

Calculate the final probability

\[P(X \geq 7) = \] (Calculate the value of P(X=7), P(X=8), P(X=9), and P(X=10) from Step 3 and sum them as per the equation in Step 4) The final probability \(P(X \geq 7)\) represents the probability of the individual getting at least 7 out of 10 correct predictions by chance alone, without having extrasensory perception (ESP).