You're viewing 1 free preview question for Maths AI HL — subscribe to unlock 14 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Probability Distributions · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQBinomial distribution and its properties10 marksPaper 3~15 min

A quality control engineer investigates the reliability of a production line for electronic components. The line is considered to be operating correctly if the proportion of defective components is at most

p = 0.05

. A random sample of

n = 20

components is selected. The number of defective components,

X

, is modelled as

X \sim B(20,\, 0.05)

. The binomial PMF is

P(X = x) = \dbinom{n}{x} p^x (1-p)^{n-x}

and the identity

\displaystyle\sum_{x=0}^{n} \binom{n}{x} a^{x} b^{n-x} = (a+b)^n

may be used without proof.

(a)

Using the PMF and the identity above, show that

E(X) = np

for

X \sim B(n, p)

. [4 marks]

(b)

The engineer shuts down the line if

X \geq 3

. Calculate the probability that the line is shut down when it is actually operating correctly. [3 marks]

(c)

In practice, components are produced sequentially on a single machine that gradually wears over a shift. A colleague suggests replacing the binomial model with one in which the probability of a defect increases linearly with the position of the component in the sequence. Evaluate whether the binomial model or the colleague's proposed modification is more appropriate for this context. Justify your answer with reference to the assumptions of the binomial distribution. [3 marks]

Solutions

Questions 2-15

Unlock these questions with Premium

That's every free question here — 1 of 15.

+14 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade