Question

A weekend TV game show called rolling a dice is running each week. For each roll, if the dice shows an odd number the participate earns 5 dollars and otherwise, he gets nothing. Each participate can roll the dice 20 times. Let X denote the number times the dice shows an odd number.

1. (2 mark) Calculate the average earning of a participate.
2. Calculate the probability that ? ⩽ 2 (show your final answer correct to four decimal places).
3. Under appropriate condition, a Binomial random variable X can be approximated by a Normal random variable Y with the same mean and standard deviation. In doing so, we need to apply a continuity correction. That is

? (? ⩽ ?) ≈ ? (? ⩽ ? + 0.5).

Calculating the probability that ? ≤ 6 by using Normal approximation with continuity correction

Answer 1

There are 3 odd numbers on a die and the probability that the die shows an odd number is 3/6=0.5

Let X denote the number times out of 20 throws, the die shows an odd number. We can say that X has a Binomial distribution with parameters, number of trials (number of throws) n=20, success probability ( the probability that the die shows an odd number) p=0.5

The probability that X=x times the die shows an odd number is

a) (2 mark) Calculate the average earning of a participate.

Let Y be the earnings. if the dice shows an odd number the participate earns 5 dollars. The total earnings in 20 rolls would be

The expected value of X is

The expected value of Y is

ans: the average earning of a participate is $50

b) (4 marks) Calculate the probability that ? ⩽ 2 (show your final answer correct to four decimal places).

the probability that ? ⩽ 2 is

ans: the probability that ? ⩽ 2 is 0.0002

c. (5 marks) Under appropriate condition, a Binomial random variable X can be approximated by a Normal random variable Y with the same mean and standard deviation. In doing so, we need to apply a continuity correction. That is

? (? ⩽ ?) ≈ ? (? ⩽ ? + 0.5).

a Binomial random variable X can be approximated by a Normal random variable Y with the same mean and standard deviation, when

are both 5 or more.

Hence, X can be approximated by a Normal random variable Y.

The expected value of X is

The standard deviation of  of X is

Using the normal approximation, we can say that Y has normal distribution with mean and standard deviation

the probability that ? ≤ 6 by using Normal approximation with continuity correction is

ans: the probability that ? ≤ 6 by using Normal approximation with continuity correction is 0.0582