Question

Two cards are drawn successively with replacement from a well shuffled deck of $52$ cards, then the mean of the number of aces is
A. $\dfrac{1}{13}$
B. $\dfrac{3}{13}$
C. $\dfrac{2}{13}$
D. None of these

Answer 1

In this question we will use binomial distribution and we will use probability formula and find the values of $P$, $q$ and $n$ after that we have to find out the mean of number of aces hence we will apply the formula for finding mean of aces to check which option is correct in the given options.
Complete step-by-step solution:
We know that in different situations the measure of uncertainty is called probability. The ratio of favourable number of outcomes to the total number of outcomes is the classical theory of probability .In statistical concept the probability is based on observations and collection of facts but in modern reference in axiomatic approach of probability we use some universal truth concepts.
Probability is the way of expressing knowledge of belief that an event will occur on chance.
Probability is the branch of math that studies patterns of chance. The idea of probability is based on observation, it describes what happens over many trials.
Basically there are three types of probabilities:
Theoretical Probability: It is based on the possible chances of something to happen.
Experimental Probability: It is based on the basis of the observations of an experiment.
Axiomatic Probability: In this probability a set of rules or axioms are set which applies to all types.
The formula of the probability of an event is:
$\text{probability}=\dfrac{\text{number of desired outcomes}}{\text{total number of favourable outcomes}}$
Or
$P(A)=\dfrac{n(A)}{n(S)}$
Now according to the question:
We have given that two cards are drawn successively with replacement from a well shuffled deck of $52$ cards hence here we will use binomial distribution.
As we know that in a deck of card number of aces is $4$
$X=$ number of aces $=4$
We will apply the formula $P(X=r)={}^{n}{{C}_{r}}{{p}^{r}}{{q}^{n-r}}$
Where $n=2$ , as two cards are drawn.
$\text{probability}=\dfrac{\text{number of desired outcomes}}{\text{total number of favourable outcomes}}$
Hence $P=\dfrac{4}{52}$
$\Rightarrow P=\dfrac{1}{13}$
We know that $q=1-P$
$\Rightarrow q=1-\dfrac{1}{13}$
$\Rightarrow q=\dfrac{13-1}{13}$
$\Rightarrow q=\dfrac{12}{13}$
Now we have to find the mean of number of aces and we know that: $mean=nP$
$\Rightarrow mean=2\times \dfrac{1}{13}$
$\Rightarrow mean=\dfrac{2}{13}$
Hence we can say that option $(3)$ is correct as the mean is $\dfrac{2}{13}$.

Note: In probability Null set $\phi$ and sample space $S$ also represent events because both are the subsets of $S$. Here $\phi$ represents an impossible event and $S$ represents a definite event. The subset $\phi$ of $S$ denotes an impossible event and the subset of $S$ of $S$itself denotes the sure event.