Binomial expectation and pass probability compared with a distribution table whose unknowns come from its total and mean
Binomial expectation and pass probability compared with a distribution table whose unknowns come from its total and mean
17 marks
A quiz has six questions, and a student passes the quiz by answering at least four of them correctly. Kiran answers each question correctly with probability \(0.4\), independently of the other questions. Let \(X\) be the number of questions that Kiran answers correctly.
- Find \(\mathrm{E}(X)\).
- Find the probability that Kiran passes the quiz.[6]
Devi takes the same quiz. The number of questions that Devi answers correctly is \(Y\), with the probability distribution given in the table, where \(a\) and \(b\) are constants.
| \(y\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|---|---|---|---|---|---|---|
| \(\mathrm{P}(Y=y)\) | \(0.12\) | \(0.25\) | \(2a+b\) | \(a+2b\) | \(3a-b\) | \(b\) | \(0.03\) |
- Show that \(2a+b=0.2\).
- Given that \(\mathrm{E}(Y)=2.3\), find the value of \(a\) and the value of \(b\).[8]
- Determine which of the two students is more likely to pass the quiz.[3]
Solution:
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Answer:(a)(i) \(\mathrm{E}(X)=2.4\) (a)(ii) \(0.1792\) (b)(i) \(2a+b=0.2\) (shown) (b)(ii) \(a=0.07,\ b=0.06\) (c) Devi, since \(0.24>0.1792\)
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