Predicting Preference: Using Binomial Distribution for a New Dairy For

Step 1 of 4

Given four judges is given 3 glasses in random order.

Our goal is:

a). We need to find the probability function for Y.

b). We need to find the probability that at least three of the four judges state a

      preference for the new formula.

c). We need to calculate the expected value of Y.

d). We need to calculate the variance of Y.

a).

Now we have to find the probability function for Y.

Let the number of judges stating a preference for the new formula is Y.

Let Y be a binomial with two random variable n and p.

The formula of the binomial is

\(n c_{y}(p)^{y}(1-p)^{n-y}\) ,where \(y=0,1,2,3,4\).

Here, Y is binomial with n = 4 and \(\mathrm{p}=\frac{1}{3}=\)P(judges choose formula B).

The probability function for Y is

\(\begin{array}{l} \mathrm{P}(\mathrm{y})=4 c y\left(\frac{1}{3}\right)^{y}\left(1-\frac{1}{3}\right)^{4-y} \\ \mathrm{P}(\mathrm{y})=4 c y\left(\frac{1}{3}\right)^{y}\left(\frac{2}{3}\right)^{4-y} \end{array}\)

Hence, \(\mathrm{P}(\mathrm{y})=4 c y\left(\frac{1}{3}\right)^{y}\left(\frac{2}{3}\right)^{4-y}\)