Question

The probability of winning on a slot machine is 5%. If a person plays the machine 500 times, find the probability of winning at least 30 times. Group of answer choices Greater than 0.60 Between 0.20 and 0.40 Between 0.01 and 0.20 Between 0.40 and 0.60 Almost 0

Answer 1

Answer:

Between 0.01 and 0.20

Step-by-step explanation:

I am going to use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$n = 500, p = 0.05$

So

$\mu = E(X) = np = 500*0.05 = 25$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{500*0.05*0.95} = 4.8734/tex] Find the probability of winning at least 30 times.Using continuity correction, this is [tex]P(X \geq 30 - 0.5) = P(X \geq 29.5)$. So this is 1 subtracted by the pvalue of Z when X = 29.5. Then

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{29.5 - 25}{4.8734}$

$Z = 0.92$

$Z = 0.92$ has a pvalue of 0.8212

1 - 0.8212 = 0.1788

So the correct option is:

Between 0.01 and 0.20