Answer to Question #340142 in Statistics and Probability for Maneo

Question

Consider the experiment of throwing a pair of dice. We are interested in the sum of the two numbers that occur. We know that X can assume the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

1) What is the probability mass function of X?

2) What is the probability that the number of dots will be at least 4 and at most 7?

3) Find the probability that the sum will be at least 7.

Answer 1

1)

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hdashline 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hdashline 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hdashline 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hdashline 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hdashline 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hdashline \end{array}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & p(x) \\ 2 & 1/36\\ 3 & 3/36 \\ 4 & 4/36 \\ 5 & 5/36 \\ 6 & 6/36 \\ 7 & 6/36 \\ 8 & 5/36 \\ 9 & 4/36 \\ 10 & 3/36 \\ 11 & 2/36 \\ 12 & 1/36 \\ \end{array}$$

2)

$$P(4\le X\le 7)=\dfrac{4+5+6+6}{36}=\dfrac{7}{12}$$

3)

$$P( X\ge 7)=\dfrac{6+6+5+4+3+2+1}{36}=\dfrac{9}{12}$$