A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Solution

Total number of outcomes = 36

The possible doublets are ( 1, 1 ), ( 2, 2 ), ( 3, 3 ), ( 4, 4 ), ( 5, 5 ), and ( 6, 6).

Let p be the probability of success, therefore,

$p = \frac{6}{36} = \frac{1}{6} So, q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6}$

Since the dice is thrown 4 times, n = 4

Let X denote the number of times of getting doublets in the experiment of throwing two dice simultaneously four times.

Therefore X can be take the values 0, 1, 2, 3, or 4.

$P (X = 0) =^{4} C_{0} p^{0} q^{4} = {(\frac{5}{6})}^{4} = \frac{625}{1296} P (X = 1) =^{4} C_{1} p^{1} q^{3} = 4 (\frac{1}{6}) {(\frac{5}{6})}^{3} = \frac{500}{1296} P (X = 2) =^{4} C_{2} p^{2} q^{2} = 6 {(\frac{1}{6})}^{2} {(\frac{5}{6})}^{2} = \frac{150}{1296} P (X = 3) =^{4} C_{3} p^{3} q = 4 {(\frac{1}{6})}^{3} (\frac{5}{6}) = \frac{20}{1296} P (X = 4) =^{4} C_{4} p^{4} q^{0} = {(\frac{1}{6})}^{4} = \frac{1}{1296}$

Thus the probability distribution is:

X	0	1	2	3	4
P(X)	$\frac{625}{1296}$	$\frac{500}{1296}$	$\frac{150}{1296}$	$\frac{20}{1296}$	$\frac{1}{1296}$

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Probability

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

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