A pair of dice is loaded. The probability that a 2 appears on the first die is 3/13 and the probability that a 4 appears on the second die is 3/13. Other outcomes for each die appear with probability 2/13. What is the probability of 6 appearing as the sum of the numbers when the two dice are rolled?
$\endgroup$2 Answers
$\begingroup$Add up the following:
- Probability of $1$ on 1st dice and $5$ on 2nd dice: $\frac{2}{13}\cdot\frac{2}{13}$
- Probability of $2$ on 1st dice and $4$ on 2nd dice: $\frac{3}{13}\cdot\frac{3}{13}$
- Probability of $3$ on 1st dice and $3$ on 2nd dice: $\frac{2}{13}\cdot\frac{2}{13}$
- Probability of $4$ on 1st dice and $2$ on 2nd dice: $\frac{2}{13}\cdot\frac{2}{13}$
- Probability of $5$ on 1st dice and $1$ on 2nd dice: $\frac{2}{13}\cdot\frac{2}{13}$
And you get $\frac{25}{169}$
$\endgroup$ 1 $\begingroup$If $P(a)P(b)$ represents $a$ appearing in the first dice, $b$ in the second,
$$P(1)P(5)+P(2)P(4)+P(3)P(3)+P(4)P(2)+P(5)P(1)$$
$$=\frac2{13}\cdot\frac2{13}+\frac3{13}\cdot\frac3{13}+\frac2{13}\cdot\frac2{13}+\frac2{13}\cdot\frac2{13}+\frac2{13}\cdot\frac2{13}$$
$$=\frac9{169}+4\cdot\frac4{169}$$
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