Probability distribution from conditional setup

A question is this type if and only if it requires constructing a complete probability distribution table for a random variable defined in a conditional probability context.

4 questions

CAIE S1 2023 November Q5
5 A red spinner has four sides labelled \(1,2,3,4\). When the spinner is spun, the score is the number on the side on which it lands. The random variable \(X\) denotes this score. The probability distribution table for \(X\) is given below.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.28\(p\)\(2 p\)\(3 p\)
  1. Show that \(p = 0.12\).
    A fair blue spinner and a fair green spinner each have four sides labelled 1, 2, 3, 4. All three spinners (red, blue and green) are spun at the same time.
  2. Find the probability that the sum of the three scores is 4 or less.
  3. Find the probability that the product of the three scores is 4 or less given that \(X\) is odd.
CAIE S1 2013 November Q7
7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.
  1. Find the probability that the two digits chosen are equal.
  2. Find the probability that one digit is a 5 and one digit is not a 5 .
  3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
  4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).
CAIE S1 2014 November Q7
7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
  2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
  3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.
CAIE S1 2019 November Q6
6 A box contains 3 red balls and 5 white balls. One ball is chosen at random from the box and is not returned to the box. A second ball is now chosen at random from the box.
  1. Find the probability that both balls chosen are red.
  2. Show that the probability that the balls chosen are of different colours is \(\frac { 15 } { 28 }\).
  3. Given that the second ball chosen is red, find the probability that the first ball chosen is red.
    The random variable \(X\) denotes the number of red balls chosen.
  4. Draw up the probability distribution table for \(X\).
  5. Find \(\operatorname { Var } ( X )\).