In a binomial distribution, the expected number of successes is calculated using which formula?

The expected number of successes is calculated using E(X) = μ = np, where n is the number of trials and p is the probability of success.

Fill in the blank: In a binomial distribution, the variance is given by V(X) = ___ where q = 1 - p.

Npq

Determine the probability of obtaining exactly 3 successes in 6 trials if the probability of success in each trial is 0.5.

The probability is calculated using the binomial formula P(X = 3) = C(6,3)(0.5)³(0.5)³, which evaluates to 0.3125.

What is the significance of the constants p and q in the context of a binomial distribution?

Significance of p and q

• p is success probability.

• q is failure probability.

• q equals 1 minus p.

If a manufacturer finds that 12% of his products are defective, what is the probability of having at most 2 defects in a sample of 10?

Fill in the blank: The probability of failure in a binomial trial is denoted by ___.

Q

How does one calculate the probability of obtaining at least 3 successes in a binomial distribution with 6 trials and a success probability of 0.5?

To find P(X ≥ 3), calculate 1 - P(X ≤ 2) using the binomial formula for 0, 1, and 2 successes.

What role does the combination formula play in calculating probabilities in a binomial distribution?

Combination formula is crucial.

• Counts successful outcomes.

• Calculates probabilities accurately.

• Essential for binomial distribution.

Calculate the probability of getting 4 or more successes in 10 trials if the success probability is 0.8.

Calculate P(X ≥ 4) = 1 - P(X ≤ 3) using the appropriate binomial probabilities.

What is the impact of increasing the number of trials n on the variance of a binomial distribution?

Increasing n will increase the variance, since V(X) = npq, where both n and q are factors that contribute to the variance.

True or False: The binomial distribution can only model situations with a constant probability of success.

True. The probability of success must remain constant for the trials to be considered a binomial distribution.

In a binomial distribution, the expected number of successes is calculated using the formula E(X) = ___, where n represents the number of trials and p represents the probability of success.

E(X) = np