Sure, I'll guide you step by step to compute the probability of getting exactly [tex]\(x = 2\)[/tex] successes in [tex]\(n = 60\)[/tex] independent trials with a success probability of [tex]\(p = 0.02\)[/tex] for each trial.
### Step 1: Understand the Binomial Formula
The probability of [tex]\(x\)[/tex] successes in [tex]\(n\)[/tex] independent Bernoulli trials with probability [tex]\(p\)[/tex] of success on each trial is given by the binomial probability formula:
[tex]\[ P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} \][/tex]
### Step 2: Define the Parameters
From the question, we have:
- [tex]\( n = 60 \)[/tex] (number of trials)
- [tex]\( p = 0.02 \)[/tex] (probability of success in each trial)
- [tex]\( x = 2 \)[/tex] (number of successes)
### Step 3: Compute the Binomial Coefficient
The binomial coefficient [tex]\(\binom{n}{x}\)[/tex] is calculated as:
[tex]\[ \binom{n}{x} = \frac{n!}{x!(n - x)!} \][/tex]
Substituting [tex]\(n = 60\)[/tex] and [tex]\(x = 2\)[/tex]:
[tex]\[ \binom{60}{2} = \frac{60!}{2! \times 58!} = \frac{60 \times 59}{2 \times 1} = 1770 \][/tex]
### Step 4: Compute [tex]\( p^x \)[/tex]
The term [tex]\( p^x \)[/tex] where [tex]\(p = 0.02\)[/tex] and [tex]\(x = 2\)[/tex] is:
[tex]\[ (0.02)^2 = 0.0004 \][/tex]
### Step 5: Compute [tex]\( (1 - p)^{n - x} \)[/tex]
The term [tex]\( (1 - p)^{n - x} \)[/tex] where [tex]\(1 - p = 0.98\)[/tex], [tex]\(n = 60\)[/tex], and [tex]\(x = 2\)[/tex] is:
[tex]\[ (0.98)^{60 - 2} = (0.98)^{58} \][/tex]
### Step 6: Combine All Terms
Using all these values in the binomial probability formula:
[tex]\[ P(X = 2) = \binom{60}{2} \times (0.02)^2 \times (0.98)^{58} \][/tex]
### Final Result
After performing all these computations, the result is approximately:
[tex]\[ P(X = 2) \approx 0.2193540452165989 \][/tex]
Thus, the probability of exactly 2 successes in 60 independent trials with a success probability of 0.02 is approximately [tex]\(0.219\)[/tex].
