Using the binomial formula for probability, Choose numbers for n, x, p and q, then answer the following:
a.Create your own word problem with the numbers you chose for n, x, p & q
b. What is the probability of success?
d. What is the number of trials?
f. What is the expected value?
To use the binomial formula for probability, we need to define the following:
n: The total number of trials or observations.
x: The number of successful outcomes in those trials.
p: The probability of success in each individual trial.
q: The probability of failure in each individual trial, where q = 1 - p.
Let's choose some numbers for each variable and answer the questions:
a. Word problem: Let's say we have n = 10 trials, where x = 5 trials are successful. The probability of success in each trial, p, is 0.3, which means there is a 30% chance of success in each individual trial. Therefore, q = 1 - p = 1 - 0.3 = 0.7, which is the probability of failure in each individual trial.
We are interested in determining the following:
b. Probability of success: Using the binomial formula, the probability of getting exactly x successful outcomes in n trials is given by the formula P(x) = (nCx) * p^x * q^(n-x), where nCx represents the number of combinations of n items taken x at a time.
In our case, P(5) = (10C5) * 0.3^5 * 0.7^(10-5).
Solving this equation will give us the probability of having exactly 5 successful outcomes.
d. Number of trials: In this word problem, n = 10. Therefore, the number of trials is 10.
f. Expected value: The expected value, denoted as E(x), represents the average number of successful outcomes you would expect in a given number of trials. It is calculated by multiplying the number of trials (n) by the probability of success in each trial (p).
In our case, E(x) = n * p = 10 * 0.3.
Therefore, the expected value is the number of trials multiplied by the probability of success in each trial.