P(X > n) or P(X ≥ n)

Find the probability that more than n trials are needed for the first success.

2 questions · Standard +0.2

5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2
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CAIE S1 2022 March Q6
12 marks Moderate -0.3
6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio \(3 : 5 : 7\) respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.
  1. Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
  2. Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.
    'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry. Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
  3. Find the probability that none of Petra's 3 chocolates has orange flavour.
  4. Find the probability that each of Petra's 3 chocolates has a different flavour.
  5. Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Edexcel FS1 2021 June Q5
18 marks Standard +0.8
  1. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
    Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.
The probability of Asha getting a red counter on any one draw is 0.07
  1. Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.
  2. Find the probability that Asha gets a red counter for the second time on her 9th draw. The probability of Davinda getting a red counter on any one draw is \(p\). Davinda draws counters until she gets \(n\) red counters. The random variable \(D\) is the number of counters Davinda draws. Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,
  3. find the value of \(p\). Jerry believes that his bag contains a smaller proportion of red counters than Asha's bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.
  4. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. Jerry gets a red counter for the first time on his 34th draw.
  5. Giving a reason for your answer, state whether or not there is evidence that Jerry's bag contains a smaller proportion of red counters than Asha's bag. Given that the probability of Jerry getting a red counter on any one draw is 0.011
  6. show that the power of the test is 0.702 to 3 significant figures.