Binomial Distribution

Step 1: Use the binomial distribution formula.
The probability of hitting the target for the 5th time at the 10th throw follows a binomial distribution. Using the binomial distribution, the correct probability is .
Thus, the correct answer is (2).

Step 1: Winning means heads occur more than tails at least 4 heads.
Step 2: Correct option given is closest to .

Step 1:
Step 2:
Step 3:

In Binomial distribution, Mean ,
Variance and the mode is
if for , the probability function is maximum.
Given and

Also,
Now ,
The distribution will have unique mode (unimodal) & the mode

Step 1: Mean np=4, variance npq=3⟹ q=(3)/(4)⟹ p=\frac14.
Step 2: Mode =lfloor (n+1)prfloor=\lfloor5·\frac14rfloor=1 or nearest integer gives 4.

(1-p)⁴=(16)/(81)⟹ 1-p=(2)/(3)⟹ p=(1)/(3) P(X=4)=p⁴=(1)/(81)

The mean of a binomial distribution is given by , and the variance is given by . We are given that the mean is 4 and the variance is 3. Using these, we can solve for and : From this, we find , and . Now, the mode of a binomial distribution is given by: Substituting and : Thus, the mode is 4.