Question

Use the binomial formula to calculate the probability of the indicated event to 3 significant digits: n = 7, p = 0.76, p(x ≤ 2) p(x ≤ 2) = 0

Answer 1

The probability of the event is 0.338. We determine the probability using the binomial formula, the expression P(x ≤ 2) can be evaluated through the binomcdf function, denoted as binomcdf(n, p, 2).

Step-by-step explanation:

In order to determine the probability using the binomial formula, the expression P(x ≤ 2) can be evaluated through the binomcdf function, denoted as binomcdf(n, p, 2).

Here, 'n' represents the number of trials, and 'p' signifies the probability of success.

With specific values provided, where n equals 7 and p is 0.76, the calculation proceeds as follows: P(x ≤ 2) = binomcdf(7, 0.76, 2) = 0.338.

This outcome indicates the probability of achieving two or fewer successes in a sequence of seven trials with a success probability of 0.76.

Employing the binomial cumulative distribution function (binomcdf) allows for a precise assessment of the likelihood of various outcomes within a specified number of trials, providing a valuable tool for probabilistic analysis in scenarios characterized by discrete, binary events.

Hence, the probability of the event is 0.338.

Answer 2

To calculate the probability of getting 2 or fewer successes in 7 trials with a success probability of 0.76, we would use the binomial formula, summing the probabilities for 0, 1, and 2 successes, or use the binomcdf function on a graphing calculator for direct calculation.

Step-by-step explanation:

The question asks us to calculate the probability of obtaining 2 or fewer successes in 7 trials (p(x ≤ 2)) in a binomial distribution where each trial has a success probability of 0.76. To find this probability, we can use the binomial formula for cumulative probabilities, which is often represented by the function binomcdf on many graphing calculators.

The calculation would typically involve summing up the probabilities of getting exactly 0, 1, and 2 successes. The formula for the binomial probability of exactly x successes in n trials is:

P(x) = (nCx) * p^x * (1-p)^(n-x)

Where:

• nCx is the combination of n items taken x at a time
• p is the probability of success on any given trial
• (1-p) is the probability of failure on any given trial

To calculate p(x ≤ 2), you would sum the probabilities for x = 0, x = 1, and x = 2 using the binomial formula. Alternatively, you can use a graphing calculator and the function binomcdf(7, 0.76, 2) to find the cumulative probability directly.