Question

Suppose a random variable, x, arises from a binomial experiment. If n = 6, and p = 0.30, find the following probabilities using technology.
a.) P (x=1)

b.) P (x=5)

c.) P (x=3)

d.) P (x=less than equal 3)

e.) P(x=greather than or equal 5)

f.) P(x=less than or equal 4)

Answer 1

Answer:

• a) P(X=1) = 0.302526

• b) P(X=5) = 0.010206

• c) P(X=3) = 0.18522

• d) P(X≤3) = 0.92953

• e) P(X≥5) = 0.010935

• f) P(X≤4) = 0.989065

Explanation:

Binomial experiments are modeled by the formula:

       $P(X=x)=C(n,x)\cdot p^x\cdot (1-p)^{(n-x)}$

Where

• P(X=x) is the probability of exactly x successes

• $C(n,x)=\dfrac{n!}{x!\cdot (n-x)!}$

• p is the probability of one success, which must be the same for every trial, and every trial must be independent of other trial.

• n is the number of trials

• 1 - p is the probability of fail

• there are only two possible outcomes for each trial: success or fail.

a.) P (x=1)

       $P(X=1)=\dfrac{6!}{1!\cdot (6-1)!}\times (0.3)^1\times(1-0.3)^{(6-1)}=0.302526$

b.) P (x=5)

     $P(X=5)=\dfrac{5!}{5!\cdot (6-5)!}\times (0.3)^5\times (1-0.3)^{(6-5)}=0.010206$

c.) P (x=3)

Using the same formula:

    $P(X=3)=0.18522$

d.) P (x less than or equal to 3)

• P(X≤3)= P(X=3) + P(X=2) + P(X=1) + P(X=0)

Also,

• P(X≤3) = 1 - P(X≥4) = 1 - P(X=4) - P(X=5) - P(X=6)

You can use either of those approaches. The result is the same.

Using the second one:

• P(X=4) = 0.059335
• P(X=5) = 0.010206
• P(X=6) = 0.000729

• P(X≤3) = 1 - 0.05935 - 0.010206 - 0.000729 = 0.92953

e.) P(x greather than or equal to 5)

• P(X≥5) = P(X=5) + P(X=6)

• P(X≥5) = 0.010206 + 0.000729 =  0.010935

f.) P(x less than or equal 4)

• P(X≤4) = 1 - P(X≥5) = 1 - P(X=5) - P(X=6)

• P(X≤4) = 1 - 0.010206 - 0.000729 = 0.989065