Conditional probability in selection

Calculate conditional probabilities or determine independence/exclusivity of events defined by selection outcomes (e.g., probability of event A given event B occurred).

2 questions · Standard +0.3

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CAIE S1 2013 November Q7
11 marks Standard +0.3
7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.
  1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
  2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
  3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
  4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.
Pre-U Pre-U 9794/1 Specimen Q12
10 marks Standard +0.3
12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.
\(x\)13579
\(\mathrm { P } ( X ) = x\)0.3\(p\)0.2\(2 p\)0.2
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
  3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
  4. Find the probability that both generate a prime number.
  5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
  6. What is the probability that the sisters are not next to each other?
  7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
  8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
  9. How many ways are there of choosing the teams if each sister is in one or other of the teams?