Probabilities for a Binomial Distribution no, P)p P for x 0, 1,2, 3,.., . (2)T,)

Question

Probabilities for a Binomial Distribution no, P)p P for x 0, 1,2, 3,.., . (2)T,) and Mean: -np ?,/np(1-p) Example: Toss a balanced coin three times and count the number of heads. HHH HHT, HTH, HTT, THH,THT, TTH, TTT Standard Deviation: T+ 2/2 23-4 Wpie lo a pt Ta ATD TH ITT b) Write the sample space c) Write the probability distribution of X. Find the number of ways that you can get two heads when you flip the coin three times using the formula above and compare to the results from part a. d) e) Use the Binomial formula to find the probability of exactly two heads. f) Find the mean of this situation. g) Find the standard deviation. 49

Explanation / Answer

Let X denote the random variable denoting the number of heads obtained when three coins are tossed simultaneously.

The parameters for this Binomial random variable are:

n = 3, p = 0.5

(a)

X can take values 0, 1, 2 and 3.

The probability distribution for this Random Variable is:

p = 3Cx*(0.5^x)*(0.5^(3-x)) ; for x = 0, 1, 2 or 3

(d) and (e)

In this case we need to compute P(X=2)

Using the formula above:

P(X=2) = 3C2*(0.5^2)*(0.5^(3-2)) = 0.375

(f) and (g)

For a binomial distribution, we have:

mean = n*p = 3*0.5 = 1.5

Standard deviation = (n*p*(1-p))^0.5 = (3*0.5*(1-0.5))^0.5 = 0.866

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