Question

An experiment of drawing a random card from an ordinary playing cards deck is done with replacing it back. This was done 10 times. Find the probability of getting 2 spades, 3 diamonds, 3 clubs and 2 hearts.
(a)0.024
(b)0.029
(c)0.020
(d)0.032

Answer 1

Hint: First we will write the number of number of spades, diamonds, club and hearts are there in a deck and then we will find the probability of each of the given cases in the question and multiply them all and then we also have to take the consideration of total number of arrangement and multiply it to get the final answer.

Complete step-by-step answer:
We know that the number of diamonds, spades, club and hearts in a deck are 13 each and total cards is 52.
Now it is given that whenever a card is drawn it is replaced back before the next card is drawn.
Hence, the probability of getting 2nd spade will be the same as for getting 1st spade.
The probability of getting 2 spades is:
$\begin{align}
  & ={{\left( \dfrac{\text{number of spades}}{\text{total number of cards}} \right)}^{2}} \\
 & ={{\left( \dfrac{13}{52} \right)}^{2}} \\
 & =0.0625...........(1) \\
\end{align}$
The probability of getting 3 diamond is:
$\begin{align}
  & ={{\left( \dfrac{\text{number of diamonds}}{\text{total number of cards}} \right)}^{3}} \\
 & ={{\left( \dfrac{13}{52} \right)}^{3}} \\
 & =0.015625..........(2) \\
\end{align}$
The probability of getting 3 club is:
$\begin{align}
  & ={{\left( \dfrac{\text{number of clubs}}{\text{total number of cards}} \right)}^{3}} \\
 & ={{\left( \dfrac{13}{52} \right)}^{3}} \\
 & =0.015625...........(3) \\
\end{align}$
The probability of getting 2 hearts is:
$\begin{align}
  & ={{\left( \dfrac{\text{number of hearts}}{\text{total number of cards}} \right)}^{2}} \\
 & ={{\left( \dfrac{13}{52} \right)}^{2}} \\
 & =0.0625...........(4) \\
\end{align}$
Now the total number of arrangements of 2 spades, 3 diamond, 3 club and 2 hearts will be,
$\dfrac{10!}{2!\times 3!\times 3!\times 2!}$
Now we will multiply it with the multiplication of all the probabilities.
Hence the probability of getting 2 spades, 3 diamond, 3 club and 2 hearts will be,
$\begin{align}
  & =\dfrac{10!}{2!\times 3!\times 3!\times 2!}\left( 1 \right)\times \left( 2 \right)\times \left( 3 \right)\times \left( 4 \right) \\
 & =\dfrac{10!}{2!\times 3!\times 3!\times 2!}0.0625\times 0.015625\times 0.0625\times 0.015625 \\
 & =0.024 \\
\end{align}$
Therefore, option (a) is correct.

Note: Students might miss the point that replacement is there and calculate the probability of each case one by one by decreasing the total number of cards. Hence, this point must be kept in mind. The formula of probability must be used properly and evaluated carefully.