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 p.

Explanation:

Step1: Understand binomial distribution concept

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

Step2: Determine range of successes

The number of successes \(k\) in \(n\) independent Bernoulli trials (coin - flips) can range from \(0\) to \(n\). Since \(n=5\), \(k\) can range from \(0\) to \(5\). So the second statement is true.

Step3: Recall independence of coin - flips

Coin - flips are independent events. The outcome of one coin - flip does not affect the outcome of another. So the third statement is true.

Step4: Analyze symmetry of binomial distribution

The binomial probability formula is \(P(X = k)=C(n,k)p^{k}(1 - p)^{n - k}\), where \(n = 5\), \(p=\frac{1}{2}\) for a fair coin. \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times 1}=10\) and \(C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5\times4}{2\times 1}=10\). So \(P(2\text{ heads}) = P(3\text{ heads})\). The fourth statement is true.

Step5: Sum of binomial probabilities

The sum of the probabilities in a binomial distribution \(\sum_{k = 0}^{n}P(X=k)=1\), not \(p\). So the fifth 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, \(P(2\text{ heads}) = P(3\text{ heads})\).