First slide

Theorems of probability

Question

One ticket is selected at random from 100 tickets numbered $00, 01, 02, \dots, 99$ . Suppose A and B are the sum and product of the digit found on the ticket, respectively. Then $P ((A = 7) / (B = 0))$ is given by

Moderate

A

2/13
B

2/19
C

1/50
D

none of these

Solution

The sum of the digits can be 7 in the following ways : 07, 16, 25, 34, 43, 52, 61, 70.
$∴ (A = 7) = {07, 16, 25, 34, 43, 52, 61, 70}$
Similarly,
$(B = 0) = {00, 01, 02, \dots, 10, 20, 30, \dots, 90}$
Thus,
$(A = 7) \cap (B = 0) = {09, 70}$
$∴ P ((A = 7) \cap (B = 0)) = \frac{2}{100}, P ((B = 0)) = \frac{19}{100}$
Hence,
$\begin{matrix} P (A = 7 ∣ B = 0) = \frac{P ((A = 7) \cap (B = 0))}{P (B = 0)} \\ = \frac{\frac{2}{100}}{\frac{19}{100}} = \frac{2}{19} \end{matrix}$

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