Mareena rolled a fair die twice. The result of the first roll is X1 and the result of the second roll is X2.
If X1 + X2 = 7, then the probability that X1 = 4 or X2 = 4 is around _____
4 Decimal Places
By the definition of conditional probability,
"P(X_1 = 4 or X_2 = 4 | X_1+X_2=7) = \\dfrac{P(X_1 = 4 \\hspace{1mm} or \\hspace{1mm} \\hspace{1mm}X_2 = 4 \\hspace{1mm} and \\hspace{1mm}P(X_1+X_2=7)}{P(X_1+X_2 = 7)}"
"= \\dfrac{P((X_1 = 4\\hspace{1mm} and\\hspace{1mm} X_1+X_2 = 7) or (X_2=4 \\hspace{1mm}and\\hspace{1mm} X_1+X_2=7)) }{P(X_1+X_2 = 7)}"
Assuming a standard 6 sided fair die,
If "X_1 = 4" , then "X_1+X_2 = 7" means "X_2 = 3" , otherwise,
If "X_2 = 4" , then "X_1 = 3"
Both outcomes are mutually exclusive with probability "\\dfrac{1}{36}" each, Hence total probability
"= \\dfrac{1}{36}+ \\dfrac{1}{36} = \\dfrac{1}{18}"
Of the 36 possible outcomes, there are 6 ways to sum the integers 1-6 to get 7:
"(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)"
and so a sum of 7 occurs "\\dfrac{6}{36} = \\dfrac{1}{6}" of the time.
Then the probability we want is:
"P(X_1 =4 or X_2 | X_1+X_2 = 7) = \\dfrac{\\dfrac{1}{18}}{\\dfrac{1}{6}} = \\dfrac{1}{3}"
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