Answer:
(a) The distribution of <em>X</em> is Binomial distribution.
(b) The probability of obtaining at least 3 oil strikes if 8 wells are dug is 0.0381.
(c) The probability that 8 or fewer wells will strike oil is 0.9876.
Step-by-step explanation:
Let random variable <em>X </em>= number of exploratory oil well drilled in a certain region should strike oil.
(a)
The probability of successful strike is, P (X) = <em>p</em> = 0.10.
The number of wells dug is, <em>n</em> = 8.
The outcome of the random variable <em>X</em> are:
- Successful strike.
- Unsuccessful strike.
The event of striking oil at one drilling location is independent of success at another location.
The random variables satisfies all the properties of a binomial random variable.
Thus, the distribution of <em>X</em> is Binomial distribution.
(b)
The probability function of a Binomial distribution is:

Compute the probability of obtaining at least 3 oil strikes if 8 wells are dug as follows:
P (X ≥ 3) = 1 - P (X < 3)
= 1 - P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)

Thus, the probability of obtaining at least 3 oil strikes if 8 wells are dug is 0.0381.
(c)
A Poisson distribution is used to approximate the Binomial distribution when the following conditions are satisfied:

The sample size is, <em>n</em> = 40.
The P (Success) = <em>p</em> = 0.10 (small)
Check the conditions as follows:

<em>n</em> = 40 > 20.
Thus, a Poisson distribution can be used to approximate the Binomial distribution of the random variable <em>X</em>.
The random variable <em>X</em> thus follows a Poisson distribution with parameter
.
The probability function of a Poisson distribution is:

Compute the probability that 8 or fewer wells will strike oil as follows:
P (X ≤ 8) = P (X = 0) + P (X = 1) + P (X = 2) + ... + P (X = 8)

Thus, the probability that 8 or fewer wells will strike oil is 0.9876.