Answer to Question #188496 in Statistics and Probability for LE5

Question #188496

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




1
Expert's answer
2021-05-07T11:38:52-0400

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}"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS