Probability of a Gambler Winning Between 62 and 70 Times: A Binomial Distribution Application

In this article, we will explore the probability of a gambler winning between 62 and 70 times out of 150 plays, given the probability of winning on each play is 0.45. We will use the binomial distribution and the normal approximation to solve this problem.

Problem Statement

A gambler plays a game of chance 150 times.

Each play has a probability of winning of 0.45. We need to find the approximate probability of winning between 62 and 70 times.

Evaluating the Binomial Distribution

The distribution in question is binomial, as it can be described as the sum of 150 independent Bernoulli trials, each with a success probability of 0.45.

Parameters of the Binomial Distribution

Number of trials, n 150, Probability of success on each trial, p 0.45.

Mean and Standard Deviation of the Binomial Distribution

The mean, μ n?p 150 ? 0.45 67.5

VarX np1 - p? 37.125 ≈ 6.09

Applying the Central Limit Theorem

Using the central limit theorem:

X_1 X_2 ... X_{150} asymp; N(μ, σ^2) N(67.5, 37.125)

Calculating the Probability Using the Normal Approximation

To find the probability of winning between 62 and 70 times, we can use the normal approximation to the binomial distribution.

Step 1: Identify the Parameters

Number of trials, n 150, Probability of winning, p 0.45.

Step 2: Calculate the Mean and Standard Deviation

Using the formulas for the mean and standard deviation:

Mean, μ n ? p 150 ? 0.45 67.5, Standard deviation, σ √(n ? p ? (1 - p)) √(150 ? 0.45 ? 0.55) ≈ √37.125 ≈ 6.09.

Step 3: Convert Values to Z-Scores

Convert 62 and 70 to z-scores:

z_{62} frac{62 - 67.5}{6.09} ≈ frac{-5.5}{6.09} ≈ -0.90 z_{70} frac{70 - 67.5}{6.09} ≈ frac{2.5}{6.09} ≈ 0.41

Step 4: Look Up the Probabilities

Using a standard normal distribution table or calculator:

Probability P(Z -0.90) ≈ 0.1841, Probability P(Z 0.41) ≈ 0.6591.

Step 5: Calculate the Probability of Winning Between 62 and 70 Times

The probability of winning between 62 and 70 times is:

P(62 X 70) P(Z 0.41) - P(Z -0.90) ≈ 0.6591 - 0.1841 0.4750

Conclusion: The approximate probability of the gambler winning between 62 and 70 times is 0.4750 or 47.50%.

Summary

We have used the normal approximation to the binomial distribution to solve for the probability of winning between 62 and 70 times out of 150 plays, with a 0.45 chance of success on each trial. The central limit theorem and properties of the standard normal distribution were instrumental in deriving this result.