Question

Баскетболист попадает мячом в корзину с вероятностью 0,4.Произведено 10 бросков мяча .Найди наивероятнейшее число попаданий.

Answer 1

Finding the Most Likely Number of Successful Shots

To find the most likely number of successful shots, we can use the binomial distribution formula. The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success.

In this case, the probability of making a shot is 0.4, and 10 shots are taken. We want to find the number of successful shots that is most likely to occur.

The formula for the binomial distribution is:

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

Where: - P(X=k) is the probability of getting exactly k successful shots, - C(n, k) is the number of combinations of n items taken k at a time, - p is the probability of success on each trial, and - n is the total number of trials.

To find the most likely number of successful shots, we need to calculate the probability for each possible number of successful shots (k) from 0 to 10, and then determine which value of k has the highest probability.

Let's calculate the probabilities for each value of k:

- For k = 0: - P(X=0) = C(10, 0) * 0.4^0 * (1-0.4)^(10-0) = 1 * 1 * 0.6^10 = 0.0060466176

- For k = 1: - P(X=1) = C(10, 1) * 0.4^1 * (1-0.4)^(10-1) = 10 * 0.4 * 0.6^9 = 0.040310784

- For k = 2: - P(X=2) = C(10, 2) * 0.4^2 * (1-0.4)^(10-2) = 45 * 0.4^2 * 0.6^8 = 0.120932352

- For k = 3: - P(X=3) = C(10, 3) * 0.4^3 * (1-0.4)^(10-3) = 120 * 0.4^3 * 0.6^7 = 0.214990848

- For k = 4: - P(X=4) = C(10, 4) * 0.4^4 * (1-0.4)^(10-4) = 210 * 0.4^4 * 0.6^6 = 0.250822656

- For k = 5: - P(X=5) = C(10, 5) * 0.4^5 * (1-0.4)^(10-5) = 252 * 0.4^5 * 0.6^5 = 0.200658125

- For k = 6: - P(X=6) = C(10, 6) * 0.4^6 * (1-0.4)^(10-6) = 210 * 0.4^6 * 0.6^4 = 0.111476736

- For k = 7: - P(X=7) = C(10, 7) * 0.4^7 * (1-0.4)^(10-7) = 120 * 0.4^7 * 0.6^3 = 0.040310784

- For k = 8: - P(X=8) = C(10, 8) * 0.4^8 * (1-0.4)^(10-8) = 45 * 0.4^8 * 0.6^2 = 0.00884736

- For k = 9: - P(X=9) = C(10, 9) * 0.4^9 * (1-0.4)^(10-9) = 10 * 0.4^9 * 0.6^1 = 0.001572864

- For k = 10: - P(X=10) = C(10, 10) * 0.4^10 * (1-0.4)^(10-10) = 1 * 0.4^10 * 0.6^0 = 0.0001048576

Now, let's compare the probabilities for each value of k:

- P(X=0) = 0.0060466176 - P(X=1) = 0.040310784 - P(X=2) = 0.120932352 - P(X=3) = 0.214990848 - P(X=4) = 0.250822656 - P(X=5) = 0.200658125 - P(X=6) = 0.111476736 - P(X=7) = 0.040310784 - P(X=8) = 0.00884736 - P(X=9) = 0.001572864 - P(X=10) = 0.0001048576

From the probabilities, we can see that the most likely number of successful shots is 4, with a probability of 0.250822656.

Therefore, based on the given information, the most likely number of successful shots out of 10 attempts is 4.