Math / Data & Statistics

QuestionFor the probability mass function $\mathrm{f}(\mathrm{x}, \mathrm{y})=\frac{x+y}{250}$ for $x=3,4,5,6,7$ and $y=3,4,5,6,7$
Find $P(X=3 \mid Y=7)$ (Write in the form of an integer)

Studdy Solution

STEP 1

1. The joint probability mass function is given by $f(x, y) = \frac{x+y}{250}$ .
2. We need to find the conditional probability $P(X=3 \mid Y=7)$ .
3. The conditional probability $P(X=x \mid Y=y)$ is given by $\frac{P(X=x, Y=y)}{P(Y=y)}$ .

STEP 2

1. Calculate $P(X=3, Y=7)$ .
2. Calculate $P(Y=7)$ .
3. Compute $P(X=3 \mid Y=7)$ .

STEP 3

Calculate $P(X=3, Y=7)$ using the joint probability mass function:
$P(X=3, Y=7) = f(3, 7) = \frac{3+7}{250} = \frac{10}{250} = \frac{1}{25}$

STEP 4

Calculate $P(Y=7)$ , which is the sum of probabilities for all $x$ values when $y=7$ :
$P(Y=7) = \sum_{x=3}^{7} f(x, 7)$
Calculate each term:
$f(3, 7) = \frac{10}{250}$ $f(4, 7) = \frac{11}{250}$ $f(5, 7) = \frac{12}{250}$ $f(6, 7) = \frac{13}{250}$ $f(7, 7) = \frac{14}{250}$
Sum these probabilities:
$P(Y=7) = \frac{10}{250} + \frac{11}{250} + \frac{12}{250} + \frac{13}{250} + \frac{14}{250} = \frac{60}{250} = \frac{6}{25}$

STEP 5

Compute $P(X=3 \mid Y=7)$ using the formula for conditional probability:
$P(X=3 \mid Y=7) = \frac{P(X=3, Y=7)}{P(Y=7)} = \frac{\frac{1}{25}}{\frac{6}{25}} = \frac{1}{6}$
Convert to an integer form:
$P(X=3 \mid Y=7) = 0$
The conditional probability $P(X=3 \mid Y=7)$ is:
$\boxed{0}$

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