At least one success

Finding minimum n such that P(X ≥ 1) exceeds a threshold, typically solved using P(X ≥ 1) = 1 - P(X = 0) = 1 - (1-p)^n.

8 questions · Moderate -0.3

2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities
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CAIE S1 2004 June Q7
11 marks Standard +0.3
7 A shop sells old video tapes, of which 1 in 5 on average are known to be damaged.
  1. A random sample of 15 tapes is taken. Find the probability that at most 2 are damaged.
  2. Find the smallest value of \(n\) if there is a probability of at least 0.85 that a random sample of \(n\) tapes contains at least one damaged tape.
  3. A random sample of 1600 tapes is taken. Use a suitable approximation to find the probability that there are at least 290 damaged tapes.
CAIE S1 2011 June Q7
10 marks Moderate -0.3
7
    1. Find the probability of getting at least one 3 when 9 fair dice are thrown.
    2. When \(n\) fair dice are thrown, the probability of getting at least one 3 is greater than 0.9. Find the smallest possible value of \(n\).
  2. A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement. The winner of the game is the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie wins the game.
CAIE S1 2019 June Q3
6 marks Moderate -0.3
3 The probability that Janice will buy an item online in any week is 0.35 . Janice does not buy more than one item online in any week.
  1. Find the probability that, in a 10 -week period, Janice buys at most 7 items online.
  2. The probability that Janice buys at least one item online in a period of \(n\) weeks is greater than 0.99 . Find the smallest possible value of \(n\).
OCR S1 2006 June Q4
7 marks Moderate -0.3
4
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
  3. The random variable \(Z\) has the distribution \(\mathrm { B } ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\).
Edexcel S2 2023 October Q1
10 marks Moderate -0.8
  1. Sam is a telephone sales representative.
For each call to a customer
  • Sam either makes a sale or does not make a sale
  • sales are made independently
Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2
  1. Find the probability that Sam makes
    1. exactly 2 sales in 14 calls,
    2. more than 3 sales in 25 calls. Sam makes \(n\) calls each day.
  2. Find the minimum value of \(n\)
    1. so that the expected number of sales each day is at least 6
    2. so that the probability of at least 1 sale in a randomly selected day exceeds 0.95
Edexcel S2 2012 January Q3
9 marks Moderate -0.8
3. The probability of a telesales representative making a sale on a customer call is 0.15 Find the probability that
  1. no sales are made in 10 calls,
  2. more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
  3. Find the least number of calls each day a representative should make to achieve this requirement.
  4. Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95
Edexcel S2 Q4
12 marks Standard +0.3
A bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is \(1 : 3 : 4\) and there are no beads of any other colour. In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.
  1. Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed. [2]
  2. Explain why this distribution would not be suitable if the beads were not replaced in the bag. [1]
  3. Find the probability that of the ten beads picked out
    1. five are blue,
    2. at least one is red. [6]
The experiment is repeated, but this time a bead is picked out and replaced \(n\) times.
  1. Find in the form \(a^n < b\), where \(a\) and \(b\) are exact fractions, the condition which \(n\) must satisfy in order to have at least a 99\% chance of picking out at least one red bead. [3]
Pre-U Pre-U 9794/3 2016 June Q4
8 marks Moderate -0.3
A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.
  1. In a packet that contains 10 sweets, find the probability that the packet contains
    1. at most 3 blue sweets, [1]
    2. exactly 3 blue sweets, [2]
    3. at least 1 blue sweet. [2]
  2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]