Answer to Question #299585 in Statistics and Probability for rk001

Question #299585

A pair of fair dice is rolled. Let X be the random variable representing the sum of the numbers that appear.







a. Construct the probability distribution of X for a pair of dice.







b. Find P (X≥ 8).







c. Find P (X≤ 7).







d. Find the probability that X takes an even value.







e. Find P (3 ≤X≤ 10)

1
Expert's answer
2022-02-21T11:11:49-0500

There are 62=366^2=36 outcomes


12345611+11+21+31+41+51+622+12+22+32+42+52+633+13+23+33+43+53+644+14+24+34+44+54+655+15+25+35+45+55+666+16+26+36+46+56+6\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1+1 & 1+2 & 1+3 & 1+4 & 1+5 & 1+6 \\ \hdashline 2 & 2+1 & 2+2 & 2+3 & 2+4 & 2+5 & 2+6 \\ \hdashline 3 & 3+1 & 3+2 & 3+3 & 3+4 & 3+5 & 3+6 \\ \hdashline 4 & 4+1 & 4+2 & 4+3 & 4+4 & 4+5 & 4+6 \\ \hdashline 5 & 5+1 & 5+2 & 5+3 & 5+4 & 5+5 & 5+6 \\ \hdashline 6 & 6+1 & 6+2 & 6+3 & 6+4 & 6+5 & 6+6 \\ \hdashline \end{array}

a.


xfP(X=x)211/36322/36433/36544/36655/36766/36855/36944/361033/361122/361211/36Sum=361\def\arraystretch{1.5} \begin{array}{c:c:c:c} & x & f & P(X=x) \\ \hline & 2 & 1 & 1/36 \\ \hdashline & 3 & 2 & 2/36 \\ \hdashline & 4 & 3 & 3/36 \\ \hdashline & 5 & 4 & 4/36 \\ \hdashline & 6 & 5 & 5/36 \\ \hdashline & 7 & 6 & 6/36 \\ \hdashline & 8 & 5 & 5/36 \\ \hdashline & 9 & 4 & 4/36 \\ \hdashline & 10 & 3 & 3/36 \\ \hdashline & 11 & 2 & 2/36 \\ \hdashline & 12 & 1 & 1/36 \\ \hdashline Sum= & & 36 & 1 \\ \hdashline \end{array}

Construct the probability distribution of X for a pair of dice.


xp(x)21/3631/1841/1251/965/3671/685/3691/9101/12111/18121/36\def\arraystretch{1.5} \begin{array}{c:c} x & p(x)\\ \hline 2 & 1/36 \\ 3 & 1/18 \\ 4 & 1/12 \\ 5 & 1/9 \\ 6 & 5/36 \\ 7 & 1/6 \\ 8 & 5/36 \\ 9 & 1/9 \\ 10 & 1/12 \\ 11 & 1/18 \\ 12 & 1/36 \\ \end{array}

b.


P(X8)=P(X=8)+P(X=9)+P(X=10)P(X\ge 8)=P(X=8)+P(X=9)+P(X=10)

+P(X=11)+P(X=12)+P(X=11)+P(X=12)

=536+436+336+236+136=512=\dfrac{5}{36}+\dfrac{4}{36}+\dfrac{3}{36}+\dfrac{2}{36}+\dfrac{1}{36}=\dfrac{5}{12}

c.


P(X7)=P(X=2)+P(X=3)+P(X=4)P(X\le 7)=P(X=2)+P(X=3)+P(X=4)

+P(X=5)+P(X=6)+P(X=7)+P(X=5)+P(X=6)+P(X=7)

=136+236+336+436+536+636=712=\dfrac{1}{36}+\dfrac{2}{36}+\dfrac{3}{36}+\dfrac{4}{36}+\dfrac{5}{36}+\dfrac{6}{36}=\dfrac{7}{12}


d.


P(X is even)=P(X=2)+P(X=4)P(X\ is\ even)=P(X=2)+P(X=4)

+P(X=6)+P(X=8)+P(X=10)+P(X=6)+P(X=8)+P(X=10)

+P(X=12)=136+336+536+536+336+P(X=12)=\dfrac{1}{36}+\dfrac{3}{36}+\dfrac{5}{36}+\dfrac{5}{36}+\dfrac{3}{36}

+136=12+\dfrac{1}{36}=\dfrac{1}{2}

e.


P(3X10)=P(X=3)+P(X=4)P(3\le X\le 10)=P(X=3)+P(X=4)

+P(X=5)+P(X=6)+P(X=7)+P(X=5)+P(X=6)+P(X=7)

+P(X=8)+P(X=9)+P(X=10)+P(X=8)+P(X=9)+P(X=10)

=236+336+436+536+636+536+436+336=\dfrac{2}{36}+\dfrac{3}{36}+\dfrac{4}{36}+\dfrac{5}{36}+\dfrac{6}{36}+\dfrac{5}{36}+\dfrac{4}{36}+\dfrac{3}{36}


=89=\dfrac{8}{9}

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