Given Information:
Probability of success = p = 25% = 0.25
Number of trials = n = 10
Required Information:
P(x < 3) = ?
Answer:
P(x < 3) = 0.5256 (option D)
Step-by-step explanation:
The given problem can be solved using binomial distribution since:
- There are n repeated trials and are independent of each other.
- There are only two possibilities: student is able to read or student is not able to read.
- The probability of success does not change with trial to trial.
The binomial distribution is given by
P(x) = ⁿCₓ pˣ (1 - p)ⁿ⁻ˣ
Where n is the number of trials, x is the variable of interest and p is the probability of success.
The variable of interest in this case is fewer than 3 students and the probability that fewer than three students are able to read is
P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2)
For P(x = 0):
Here we have x = 0, n = 10 and p = 0.25
P(x = 0) = ¹⁰C₀(0.25⁰)(1 - 0.25)¹⁰⁻⁰
P(x = 0) = (1)(0.25⁰)(0.75)¹⁰
P(x = 0) = 0.0563
For P(x = 1):
Here we have x = 1, n = 10 and p = 0.25
P(x = 1) = ¹⁰C₁(0.25¹)(1 - 0.25)¹⁰⁻¹
P(x = 1) = (10)(0.25¹)(0.75)⁹
P(x = 1) = 0.1877
For P(x = 2):
Here we have x = 2, n = 10 and p = 0.25
P(x = 2) = ¹⁰C₂(0.25²)(1 - 0.25)¹⁰⁻²
P(x = 2) = (45)(0.25²)(0.75)⁸
P(x = 2) = 0.28157
Finally,
P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2)
P(x < 3) = 0.0563 + 0.1877 + 0.28157
P(x < 3) = 0.52557
Rounding off yields
P(x < 3) = 0.5256
Therefore, the probability that fewer than three of the students are able to read is 0.5256
