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)

a) P(X=1) = 0.302526b) P(X=5) = 0.010206c) P(X=3) = 0.18522d) P(X≤3) = 0.92953e) P(X≥5) = 0.010935f) P(X≤4) = 0.989065

Explanation:

Binomial experiments are modeled by the formula:

       

Where

P(X=x) is the probability of exactly x successesp 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 trials1 - p is the probability of failthere are only two possible outcomes for each trial: success or fail.

a.) P (x=1)

       

b.) P (x=5)

     

c.) P (x=3)

Using the same formula:

   

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

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