You draw one card from a 52 card deck. Then the card is replaced in the deck and the deck is shuffled, and you draw again. Find the probability of drawing an eight the first time and a diamond the second time.

Solution

Y o u d r a w o n e c a r d f ro m a ​ 52 c a r d d ec k . T h e n t h e c a r d i s re pl a ce d in t h e d ec k an d t h e d ec k i s ​ s h u ff l e d, an d yo u d r a w a g ain . F in d t h e p ro babi l i t y o f d r a w in g an e i g h t t h e f i rs t t im e an d a d iam o n d t h e seco n d t im e .

\frac{1}{52}

Decimal

0.01923 \dots

Solution steps

Find the probability of drawing an 8 the first time:

\frac{1}{13}

Find the probability of a diamond the second time:

\frac{1}{4}

Since the events are independent, Use the formula

P (X_{1} \land X_{2} \land \dots \land X_{k}) = P (X_{1}) \cdot P (X_{2}) \cdot \dots \cdot P (X_{k})

= \frac{1}{13} \cdot \frac{1}{4}

\frac{1}{13} \cdot \frac{1}{4} = \frac{1}{52}

= \frac{1}{52}