Question

A pair of dice is rolled 180 times. What is the probability that a total of six occurs
a) At least 30 times b)between 33 and 42 times inclusive. Use Normal approximation to Binomial distribution

Answer 1

Answer on Question #45518— Math — Statistics and Probability

A pair of dice is rolled 180 times. What is the probability that a total of six occurs a) At least 30 times b)between 33 and 42 times inclusive. Use Normal approximation to Binomial distribution.

Solution

To get six in a roll you have probability ((1,5)+(2,4)+(3,3)+(4,2)+(5,1))/($\mbox{total }6^{2}$ possibilities):

$p=\frac{5}{36}$

So, if X = number of times a total occurs out of n=180 rolls then $X\sim B(180,5/36)$.

Now Let $Y\sim N(\mu,\sigma^{2})$ be normal approximation of X

$\mu=np=180\cdot\frac{5}{36}=25,\qquad\sigma^{2}=np(1-p)=180\cdot\frac{5}{36}\cdot\frac{31}{36}\approx 21.5$

So we have

$X\sim N(25,21.5)$

Now, using tables, we can easily find: a. $X\geq 30$ which is the same $X\geq 29.5$ (continuity correction).

$P(X\geq 29.5)\approx 0.417106$

b. with continuity corrections: $32.5\leq X\leq 42.5$:

$P(32.5\leq X\leq 42.5)\approx 0.363606$

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