Question

Explain why P(A|D) and P(D|A) from the table below are not equal.

Answer 1

Answer:

Sample Response: The two conditional probabilities are not equal because each has different given events. P(A|D) has event D as its given event, resulting in 2/10 for a probability. P(D|A) has event A as its given event, resulting in 2/8 for a probability.

Answer 2

Answer:

The Probability that Event A and  Event D occur is equal to the probability Event A occurs times the probability that Event D occurs, given that A has occurred.

$P(A\cap D)=P(A)\cdot P(D/A)$

We can find the values of $P(A/D)$ and $P(D/A)$ using the above form formula.

$P(D/A)=\frac{P(A\cap D)}{P(A)}$ ;

$P(A/D)=\frac{P(D\cap A)}{P(D)}$

From the given table, we have the values of P(A), P(D), $P(D\cap A)$ and $P(A\cap D)$.

Since, Probability=$\frac{The number of wanted outcomes }{the number of possible outcomes}$

∴$P(A)=\frac{8}{17}$, $P(D)=\frac{10}{17}$, $P(A\cap D)=\frac{2}{17}$ and $P(D\cap A)=\frac{2}{17}$

Now, putting these values in above formula we get,

$P(D/A)=\frac{\frac{2}{17}} {\frac{8}{17}}$

$P(D/A)=\frac{2} {8}=\frac{1}{4}$

$P(D/A)= \frac{1}{4}$.

$P(A/D)=\frac{\frac{2}{17}} {\frac{10}{17}}=\frac{2}{10}$

$P(A/D)=\frac{1}{5}$

As, you can see above that the values of P(A|D) and P(D|A) are not equal.