IB Mathematics SL 4.8 Binomial distribution. Mean and variance-AI SL Paper 1- Exam Style Questions- New Syllabus
Question
Most appropriate topic codes (IB Mathematics: applications and interpretation):
• SL 4.8: Calculation of binomial probabilities using technology — part (b)
▶️ Answer/Explanation
(a)
One of the following:
• The number of trials is fixed (5 children).
• There are only two outcomes for each child: brown eyes or not brown eyes.
• The probability of brown eyes is constant for each child (0.75).
\(\boxed{\text{Fixed number of trials}}\) or equivalent.
(b)(i)
Let \(X \sim B(5, 0.75)\).
\(P(X = 3) = \binom{5}{3} (0.75)^3 (0.25)^2\)
\(= 10 \times 0.421875 \times 0.0625 = 0.263671875\)
\(\approx 0.264 \quad \text{(to 3 s.f.)}\)
\(\boxed{0.264}\)
(b)(ii)
\(P(X \geq 4) = P(X = 4) + P(X = 5)\)
\(P(X = 4) = \binom{5}{4} (0.75)^4 (0.25)^1 = 5 \times 0.31640625 \times 0.25 = 0.3955078125\)
\(P(X = 5) = (0.75)^5 = 0.2373046875\)
\(P(X \geq 4) = 0.3955078125 + 0.2373046875 = 0.6328125\)
\(\approx 0.633 \quad \text{(to 3 s.f.)}\)
\(\boxed{0.633}\)