Question

select all statements below that are true about the binomial distribution shown on the right.
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 heads)=p(3 heads).
the sum of the probabilities shown in the binomial distribution is 1.

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, \(P(2\text{ heads}) = P(3\text{ heads})\)