(a)
8

Probability of failing: 0.2

Probability of passing: \( 1 – 0.2 = 0.8 \)

Number of people: \( n = 10 \)

Expected passes: \( E(X) = n \times p = 10 \times 0.8 = 8 \)

Result: 8 [1]

(b)
0.0881

Binomial distribution: \( n = 10 \), \( p_{\text{fail}} = 0.2 \), \( p_{\text{pass}} = 0.8 \)

Probability of 4 failures: \( P(X = 4) = \binom{10}{4} \times (0.2)^4 \times (0.8)^6 \)

\( \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \)

\( (0.2)^4 = 0.0016 \)

\( (0.8)^6 \approx 0.262144 \)

\( P(X = 4) = 210 \times 0.0016 \times 0.262144 \approx 0.088080384 \)

Rounded to 4 decimal places: 0.0881

Result: 0.0881 [2]

(c)
0.121

Fewer than 7 passes: \( P(\text{passes} < 7) = P(\text{failures} \geq 4) \)

\( P(\text{failures} \geq 4) = 1 – P(\text{failures} \leq 3) \)

\( P(X = k) = \binom{10}{k} \times (0.2)^k \times (0.8)^{10-k} \)

\( P(X = 0) = 1 \times 1 \times 0.1073741824 \approx 0.1073741824 \)

\( P(X = 1) = 10 \times 0.2 \times 0.134217728 \approx 0.268435456 \)

\( P(X = 2) = 45 \times 0.04 \times 0.16777216 \approx 0.301989888 \)

\( P(X = 3) = 120 \times 0.008 \times 0.2097152 \approx 0.201326592 \)

\( P(X \leq 3) \approx 0.1073741824 + 0.268435456 + 0.301989888 + 0.201326592 \approx 0.8791261184 \)

\( P(X \geq 4) = 1 – 0.8791261184 \approx 0.1208738816 \)

Rounded to 3 decimal places: 0.121

Result: 0.121 [3]