Binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.
Question 1: If a coin is tossed 3 times, then determine the probability of getting exactly two heads.
X be the random variable for number of heads
The formula for the required probability is given by: P (X = x) = nCx px qn-x
Here, n = 3, x = 2, p =1/2, q = 1/2
⇒ P (X = 2) = 3C2 (1/2)2 (1/2)3-2
⇒ P (X = 2) = 3 (1/2)3
⇒ P(X = 2) = 3/8
Question 2: A pair of dice is rolled 5 times. If getting the product of 6 is considered a success. Find the probability of getting at least 4 successes.
Here, n = 5
p is the probability of getting product 6
p = 4 /36 = 1/9
q = 1 - p = 8/9
⇒ P(X ≥ 4) = P(X = 4) + P(X = 5)
⇒ P(X ≥ 4) = 5C4 (1/9)4 (8/9)5-4+ 5C5 (1/9)5 (8/9)5-5
⇒ P(X ≥ 4) = 5(1/9)4 (8/9) + (1/9)5
Question 3: If the number of trials of a certain binomial distribution is 225 and the probability of success is 0.36. Find the standard deviation.
The formula of the standard deviation of binomial distribution is given by: σ = √(npq)
Here, n = 225, p = 0.36 and q = 0.64
⇒ σ = √(225× 0.36 × 0.64)
⇒ σ = √51.84
⇒ σ = 7.2
Question 4: The mean and the standard deviation of a binomial distribution are 100 and 5. Find the binomial distribution.
The formula of the standard deviation and mean of binomial distribution is given by:
σ = √(npq), Mean = np
σ = √(Mean × q)
Here, mean = 100 and σ = 5
5 = √(100 × q)
⇒ 25 = 100q
⇒ q = 1/4
Now, p = 1 - q = 1 - 1/4 = 3/4
By mean formula
n = 400 / 3
⇒ n = 133 (approx.)
The binomial distribution is:
P (X = x) = 133Cx (3/4)x(1/4)133-x
Question 5: Find the mean if the number of good pens is 20 and the probability of a good pen is 0.8.
The formula of mean in binomial distribution is given by:
Mean = np
Here, n = 20 and p = 0.8
⇒ Mean = 20 × 0.8
⇒ Mean = 16
Question 6: If a coin is tossed 4 times. Find the probability that a tail appears an odd number of times.
X be the random variable that tails appear.
Here,
n = 4, p = (1/2), q = 1 - p = 1/2, x = 1, 3 (odd times)
⇒ P (X = odd) = P(X = 1) + P(X = 3)
⇒ P (X = odd) = 3C1 (1/2)1 (1/2)3 - 1 + 3C3 (1/2)3 (1/2)3 - 3
⇒ P (X = odd) = 3(1/2)3 + (1/2)3
⇒ P (X = odd) = 4 / 8
⇒ P (X = odd) = 1/2
Question 7: The probability of a man hitting the target is 1/4. If he fires 3 times, what is the probability of his hitting the target at least once?
X be the random variable for man hitting target.
Here,
n = 3, p = (1/4), q = 1 - p = 3/4, x = 1, 2, 3 (hitting at least once)
Now, P (X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3)
⇒ P (X ≥ 1) = 3C1 (1/4)1 (3/4)3 - 1 + 3C2 (1/4)2 (3/4)3 - 2 + 3C3 (1/4)3 (3/4)3 - 3
⇒ P (X ≥ 1) = 3(1/4) (9/16) + 3(1/16) (3/4) + (1/4)3 (3/4)0
⇒ P (X ≥ 1) = 27/64 + 9/64 + 1/64
⇒ P (X ≥ 1) = 37/64
⇒ P (X ≥ 1) = 0.578
Question 8: The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students at the university none will graduate.
X be the random variable for student will graduate.
Here,
n = 3, p = 0.4, q = 1 - p = 0.6, x = 0 (none of student will graduate)
⇒ P (X = 0) = 3C0 (0.4)0 (0.6)3 - 0
⇒ P(X = 0) = (0.6)3
⇒ P(X = 0) = 0.216
Question 9: If the mean and variance of a binomial distribution are 8 and 2, then find P(X ≥ 1).
The formula for the mean and variance in the binomial distribution is given by:
Mean = np, Variance = npq
Variance = Mean × q
⇒ 2 = 8q
⇒ q = 2 / 8 = 0.25
Now, p = 1 - 0.25 = 0.75
n = Mean /p
⇒ n = 8 / 0.75
⇒ n = 10 (approx.)
Thus , P (X ≥ 1) = 1 − P (X=0)
⇒ P (X ≥ 1) = 1 - 10C0 (0.75)0 (0.25)10
⇒ P (X ≥ 1) = 1 − (0.25)10
⇒ P (X ≥ 1) = 1−9.5367×10 -7
⇒ P (X ≥ 1) ≈ 0.999999
Question 10: Find the variance of the binomial distribution if the mean is 40 and the probability of failure is 0.1.
The formula to find variance of binomial distribution is given by:
Variance = npq = Mean × q
⇒ Variance = 40 × 0.1
⇒ Variance = 4
Practice Problems
Q1: If a coin is tossed 5 times, find the probability of getting at most 2 heads.
Q2: A pair of dice is thrown 7 times. If getting a total of 11 is considered a success, what is the probability of getting at least 3 successes?
Q3: If the mean of the binomial distribution is 20 and the number of observations is 30. Find the variance and standard deviation.
Q4: Find the mean of the binomial distribution given that the number of trials is 80 and the probability of success is 0.45.
Q5: Calculate the variance of the binomial distribution given that the number of trials is 200 and the probability of success is 0.8.
Q6: If the mean and variance of the binomial distribution are 45 and 30. Find the binomial distribution.
Q7: If the mean and variance of a binomial distribution are 6 and 2, then find P(X>3).
Q8: The probability of a boy hitting a target is 0.2. How many times must he fire so that the probability of his hitting the target at least once is greater than 2/3?
Q9: Find the probability distribution of the number of heads when 5 coins are tossed.
Q10: There are 20 machines, and the chance of any one of them being out of service is 0.025. Determine the probability that exactly three machines will be out of service on the same day.
