Final answer for the Magician Competition Probability

What is the probability for the magician to win the competition provided that he does not have magical abilities?

Enter value with 4 digits after decimal point.

Assume that 50 magicians participate in the competition and try to prove that they have magical abilities. What is the probability that at least one magician wins?

Enter answer with 2 digits after decimal point.

Final answer:

The probability for the magician to win the competition without magical abilities is approximately 0.0004883. The probability that at least one magician wins in a competition with 50 participants is approximately 0.0238.

Explanation:

Probability in a Magician Competition

In this magician competition, the organizers are testing the null hypothesis that the probability of guessing the coin toss correctly is 1/2. They are comparing this against the alternative hypothesis that the probability is larger than 1/2.

a) Probability for the magician to win the competition without magical abilities:

To calculate this probability, we can use the binomial distribution formula:

1. Number of trials (n) = 11 (as the coin is tossed 11 times)
2. Probability of success (p) = 1/2 (as the null hypothesis assumes the probability of guessing correctly is 1/2)
3. Number of successes (k) = 0 (as the magician does not have magical abilities)

Using these values, we can calculate the probability using the formula:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Substituting the values, we get:

P(X = 0) = C(11, 0) * (1/2)^0 * (1 - 1/2)^(11-0)

Simplifying further, we get:

P(X = 0) = 1 * 1 * (1/2)^11

Calculating this, we find that the probability for the magician to win the competition without magical abilities is approximately 0.0004883.

b) Probability that at least one magician wins in a competition with 50 participants:

To calculate this probability, we can use the complement rule and the binomial distribution formula:

1. Number of trials (n) = 50 (as there are 50 participants)
2. Probability of success (p) = 0.0004883 (as calculated in part a)
3. Number of successes (k) = 1 (as we want to calculate the probability of at least one magician winning)

Using these values, we can calculate the probability using the formula:

P(X >= k) = 1 - P(X < k)

Substituting the values, we get:

P(X >= 1) = 1 - P(X < 1)

P(X >= 1) = 1 - P(X = 0)

P(X >= 1) = 1 - 0.0004883

Calculating this, we find that the probability that at least one magician wins in a competition with 50 participants is approximately 0.0238.