Question

Ravi and Rashmi are each holding $2$ red cards and $2$ black cards (all four red and all four black cards are identical). Ravi picks a card at random from Rashmi, and then Rashmi picks a card at random from Ravi. This process is repeated a second time. Let $p$ be the probability that both have all $4$ cards of the same colour. Then $p$ satisfies

• A. $p\le 5\%$
• B. $5\%<p\le 10\%$
• C. $10\%<p\le 15\%$
• D. $15\%<p$

Answer 1

The correct option is A $p\le 5\%$

There are two possible cases,
Either Ravi will have all the Red or all the black.
For having all the red, Ravi has to take red from Rashmi then Rashmi has to take black from Ravi and again do the same.
The probability of doing it
$P=2(\frac{2}{4}\times \frac{2}{5}\times \frac{1}{4}\times \frac{1}{5})=\frac{2}{100}=2\%$

Alternative

There are two possible cases,
Either Ravi will have all the Red or all the black.
For Ravi having all the red,
Initially Both have 2 Red and 2 Black card
{RRBB BBRR}
Ravi pick 1 Red card in his first pick from rashmi pack
i.e {RRRBB BBR}
Rashmi pick 1 black card in her first pick from Ravi pack
i.e, {RRRB BBBR}
Again Ravi pick 1 Red in his second pick from rashmi pack
i.e, {RRRRB BBB}
Rashmi pick 1 black card in her second pick from Ravi pack
i.e, {RRRR BBBB}
All picks are independent
Therefore required probability=$P=2(\frac{2}{4}\times \frac{2}{5}\times \frac{1}{4}\times \frac{1}{5})=\frac{2}{100}=2\%$