# Joe is playing an arcade game and has 8 chances to knock down the moving ducks. A person playing this game hits the moving duck 78% of the time.What is the probability that Joe hits the 4th duck?What is the probability that the first duck Joe hits is not until his fourth attempt?What is the probability that Joe will hit 5 ducks?Six ducks at most?At least 4 ducks?What is the probability that he will hit between 4 and 7 ducks inclusive?What is the probability that he will hit at leat one duck?

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Joe is playing an arcade game and has 8 chances to knock down the moving ducks. A person playing this game hits the moving duck 78% of the time.

What is the probability that Joe hits the 4th duck?

What is the probability that the first duck Joe hits is not until his fourth attempt?

What is the probability that Joe will hit 5 ducks?

Six ducks at most?

At least 4 ducks?

What is the probability that he will hit between 4 and 7 ducks inclusive?

What is the probability that he will hit at leat one duck?

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Step 1

1.

Find the probability that Joe hits the fourth duck:

A person playing the Arcade game hits the moving duck 78% of the time.

That is, the probability of hitting the moving duck is p = 0.78.

The probability of not hitting the moving duck is q = 1 – p = 0.22.

The number of chances to knock down the moving ducks is n = 8.

Here, hitting the ducks in 8 chances or hits will not be dependent on the other, imp-lying independence among each other. As a result, the 8 chances or hits may be considered as independent trials.

Here, all the 8 hits have a probability of 0.78 of hitting the duck. That is, the rate of success (hitting the duck) is constant. Hence, the rate of failure (not hitting the duck) is also constant.

Therefore, the probability of hitting any duck will be 0.78.

Thus, the probability that Joe hits the fourth duck is 0.78.

Step 2

Introduction to geometric distribution:

Consider a random experiment involving repeated n independent Bernoulli trials until a success is obtained.

Moreover, the probability of getting a success in each n independent trial, p, remains a constant for all the trials. Denote the probability of failure as q. As success and failure are mutually exclusive, q = 1 – p.

Let the random variable X denote the number of trails before the first success. Thus, X can take any of the values 0,1,2,….n.

Then, the probability distribution of X is a Geometric distribution with parameter (p) and the probability mass function (pmf) of X, that is, of a geometric random variable, is given as:

Step 3

2.

Find the probability that the first duck Joe hits is not until the fourth attempt:

Consider the event of hitting the moving duck as a “success”.

Consider X as the number of chances used before hitting the first moving duck, Then, X has a geometric distribution with parameter (p = 0.78) and the pmf of X is

P(x) = (078)*(0.22)x

It is given that the first duck Joe hits...

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