Multiple independent trials

Questions involving repeated independent trials with the same probability structure, calculating probabilities of specific sequences or patterns.

6 questions

CAIE S1 Specimen Q7
7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.
  1. Find the probability of throwing a 3 .
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  2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
  3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.
OCR MEI S1 Q2
2 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.
  1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
  2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
  3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.
OCR MEI S1 2016 June Q2
2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.
  • \(\mathrm { P } (\) Wins \() = 0.5\)
  • \(\mathrm { P } (\) Draws \() = 0.3\)
  • \(\mathrm { P } (\) Loses \() = 0.2\)
The outcomes of the 3 matches are independent.
  1. Find the probability that Team A does not lose in any of the 3 matches.
  2. Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
  3. Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.
AQA S1 2007 January Q5
5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are \(0.6,0.7\) and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:
  1. none of the three cyclists takes part;
  2. Fabio is the only one of the three cyclists to take part;
  3. exactly one of the three cyclists takes part;
  4. either one or two of the three cyclists take part.
AQA S1 2010 January Q4
4 Each school-day morning, three students, Rita, Said and Ting, travel independently from their homes to the same school by one of three methods: walk, cycle or bus. The table shows the probabilities of their independent daily choices.
\cline { 2 - 4 } \multicolumn{1}{c|}{}WalkCycleBus
Rita0.650.100.25
Said0.400.450.15
Ting0.250.550.20
  1. Calculate the probability that, on any given school-day morning:
    1. all 3 students walk to school;
    2. only Rita travels by bus to school;
    3. at least 2 of the 3 students cycle to school.
  2. Ursula, a friend of Rita, never travels to school by bus. The probability that: Ursula walks to school when Rita walks to school is 0.9 ; Ursula cycles to school when Rita cycles to school is 0.7 . Calculate the probability that, on any given school-day morning, Rita and Ursula travel to school by:
    1. the same method;
    2. different methods.
AQA AS Paper 2 2022 June Q14
2 marks
14 Yingtai visits her local gym regularly. After each visit she chooses one item to eat from the gym's cafe.
This could be an apple, a banana or a piece of cake.
She chooses the item independently each time.
The probability that Yingtai chooses each of these items on any visit is given by: $$\begin{aligned} \mathrm { P } ( \text { Apple } ) & = 0.2
\mathrm { P } ( \text { Banana } ) & = 0.35
\mathrm { P } ( \text { Cake } ) & = 0.45 \end{aligned}$$ For any four randomly selected visits to the gym, find the probability that Yingtai chose: 14
  1. at least one banana.
    [0pt] [2 marks]
    14
  2. the same item each time.
    14
  3. apple twice and cake twice