Chapter 8: Q. 8.8 (page 394)

8.8. On each bet, a gambler loses $1$ with probability $. 7$ , loses $2$ with probability $. 2$ , or wins $10$ with probability $. 1$ . Approximate the probability that the gambler will be losing after his first $100$ bets.

Short Answer

Expert verified

The probability is $. 6103$ .

Step by step solution

Given information

A gambler loses $1$ with probability $. 7$ , loses $2$ with probability $. 2$ , or wins $10$ with probability $. 1$ .

Explanation

Let $X_{i}$ denotes the gambler's winnings on $i$ th bet. Then, discrete random variable, takes the values, $- 1, - 2$ and $10 .$

Probabilities:

$P \{X_{i} = - 1\} = . 7, P \{X_{i} = - 2\} = . 2$ ,

And, $P \{X_{i} = 10\} = . 1$
Hence, the value of $X_{i}$ is:
$μ = E [X] = (- 1) (. 7) + (- 2) (. 2) + 10 (. 1) = - . 1 E [X^{2}] = (- 1)^{2} (. 7) + (- 2)^{2} (. 2) + 10^{2} (. 1) = 11.5$
The variance of $X_{i}$ is
$σ^{2} = Var (X_{i}) = 11.5 - (- . 1)^{2} = 11.49$

Explanation

Let, gambler's first $100$ bets:
$X = X_{1} + X_{2} + \dots + X_{100}$
The probability that the gambler will be losing after his first $100$ bets is:
$P {X < 0}$
Using the central limit theorem:
$P {X < 0} = P \{\frac{X - 100 μ}{10 σ} < \frac{(0 - . 5) - 100 μ}{10 σ}\} \approx Φ (\frac{- . 5 - 100 μ}{10 σ}) = Φ (\frac{- . 5 - 100 (- . 1)}{10 \sqrt{11.49}}) = Φ (. 28) = . 6103$

Hence, the probability is $. 6103$

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8.8. On each bet, a gambler loses $1$ with probability $. 7$ , loses $2$ with probability $. 2$ , or wins $10$ with probability $. 1$ . Approximate the probability that the gambler will be losing after his first $100$ bets.

Short Answer

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8.8. On each bet, a gambler loses 1 with probability.7, loses 2with probability .2, or wins 10with probability .1. Approximate the probability that the gambler will be losing after his first 100 bets.

Short Answer

Step by step solution

Given information

Explanation

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Study anywhere. Anytime. Across all devices.

8.8. On each bet, a gambler loses $1$ with probability $. 7$ , loses $2$ with probability $. 2$ , or wins $10$ with probability $. 1$ . Approximate the probability that the gambler will be losing after his first $100$ bets.