Two-stage binomial problems

Problems where a binomial outcome determines parameters for a second binomial experiment.

3 questions · Standard +0.5

2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities
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CAIE S1 2017 November Q4
7 marks Standard +0.8
4 A fair tetrahedral die has faces numbered \(1,2,3,4\). A coin is biased so that the probability of showing a head when thrown is \(\frac { 1 } { 3 }\). The die is thrown once and the number \(n\) that it lands on is noted. The biased coin is then thrown \(n\) times. So, for example, if the die lands on 3 , the coin is thrown 3 times.
  1. Find the probability that the die lands on 4 and the number of times the coin shows heads is 2 .
  2. Find the probability that the die lands on 3 and the number of times the coin shows heads is 3 .
  3. Find the probability that the number the die lands on is the same as the number of times the coin shows heads.
Pre-U Pre-U 9794/3 2015 June Q5
12 marks Standard +0.3
5 A garden centre grows a particular variety of plant for sale. They sow 3 seeds in each pot and there are 6 pots in a tray. The probability that a seed germinates is 0.7 , independently of any other seeds.
  1. State the probability distribution of the number of seeds in a pot that germinate.
  2. Find the probability that, in a randomly chosen pot,
    1. exactly 2 seeds germinate,
    2. at least 1 seed germinates. After the seeds have germinated and become seedlings, some are removed (and discarded) so that there remains at most 1 seedling per pot.
    3. Write out the probability distribution of the number of seedlings per pot that remain.
    4. Find the probability that there is a seedling in every one of the 6 pots in a randomly chosen tray.
OCR S1 2013 June Q7
11 marks Standard +0.3
In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows. • If \(X < 2\), the batch is accepted. • If \(X > 2\), the batch is rejected. • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted. It is given that 5\% of mugs are defective.
    1. Find the probability that the batch is rejected after just the first sample is checked. [3]
    2. Show that the probability that the batch is rejected is 0.327, correct to 3 significant figures. [5]
  2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked. [3]