   Chapter 8.5, Problem 22E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In the following problems, use a normal approximation to the binomial.Gambling If a die is rolled 300 times, what is the probability that a 6 will occur at least 55 times?

To determine

To calculate: The probability of occurring 6 on a die at least 55 times if it is rolled 300 times.

Explanation

Given Information:

For at least 55 successes in n=300 trails of occurring 6.

Formula used:

For a binomial distribution if, np5 and nq5 then the binomial distribution can be approximated accurately, with mean μ=np and standard deviation σ=npq.

Where n the number of trials, p is the probability of success, and q is the probability of failure q=1p.

And to convert the x values of binomial distribution in use z score use, z=xμσ,

The continuity correction to convert the binomial range of values to the corresponding interval of normal distribution values for any integer x is [x0.5,x+0.5].

Calculation:

Consider a dice is rolled n=300 times and the probability of occurring 6 is i.e. probability of occurring 6 out of total 6 sample spaces p=16,

μ=npμ=300(16)μ=50

And the probability of failure is q=1p,

q=116q=616q=56

And the standard deviation is,

σ=npqσ=300(16)(56)σ=41

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