select all statements below that are true about the binomial distribution shown on the right.\nthe bar for…

select all statements below that are true about the binomial distribution shown on the right.\nthe bar for any number k represents the probability of getting k successes in 5 flips.\nthe number of successes, k, can range from 0 (no success) to 5 (all successes).\neach coin - flip is independent. it is not affected by any other coin - flip.\nfor 5 coin flips, p(2 heads)=p(3 heads).\nthe sum of the probabilities shown in the binomial distribution is 1.

select all statements below that are true about the binomial distribution shown on the right.\nthe bar for any number k represents the probability of getting k successes in 5 flips.\nthe number of successes, k, can range from 0 (no success) to 5 (all successes).\neach coin - flip is independent. it is not affected by any other coin - flip.\nfor 5 coin flips, p(2 heads)=p(3 heads).\nthe sum of the probabilities shown in the binomial distribution is 1.

Answer

Explanation:

Step1: Analyze binomial distribution concept

In a binomial distribution of coin - flips, the bar for (k) represents the probability of (k) successes in (n) flips. Here (n = 5), so the bar for any number (k) represents the probability of getting (k) successes in 5 flips. This statement is true.

Step2: Determine range of successes

The number of successes (k) in (n) independent Bernoulli trials (coin - flips) can range from (0) (no successes) to (n) (all successes). Since (n = 5), (k) can range from (0) to (5). This statement is true.

Step3: Check independence of coin - flips

Coin - flips are independent events. The outcome of one coin - flip does not affect the outcome of any other coin - flip. This statement is true.

Step4: Compare probabilities

The probability mass function of a binomial distribution is (P(X=k)=C(n,k)p^{k}(1 - p)^{n - k}), where (n = 5) and (p=\frac{1}{2}) for a fair coin. (P(2\text{ heads})=C(5,2)(\frac{1}{2})^{2}(1-\frac{1}{2})^{3}=\frac{5!}{2!(5 - 2)!}\times(\frac{1}{2})^{5}=10\times\frac{1}{32}), and (P(3\text{ heads})=C(5,3)(\frac{1}{2})^{3}(1 - \frac{1}{2})^{2}=\frac{5!}{3!(5 - 3)!}\times(\frac{1}{2})^{5}=10\times\frac{1}{32}). So (P(2\text{ heads})=P(3\text{ heads})), this statement is true.

Step5: Sum of probabilities

The sum of the probabilities in a probability distribution (including a binomial distribution) is (1), not (p). This statement is false.

Answer:

The bar for any number (k) represents the probability of getting (k) successes in 5 flips; The number of successes, (k), can range from 0 (no success) to 5 (all successes); Each coin - flip is independent. It is not affected by any other coin - flip; For 5 coin - flips, (P(2\text{ heads}) = P(3\text{ heads}))