Sum or combination of independent binomial values

Find probabilities involving the sum or combined outcomes of multiple independent values drawn from the same binomial distribution (e.g., sum of two values equals a target).

4 questions

CAIE S1 2016 November Q2
2 A fair triangular spinner has three sides numbered 1, 2, 3. When the spinner is spun, the score is the number of the side on which it lands. The spinner is spun four times.
  1. Find the probability that at least two of the scores are 3 .
  2. Find the probability that the sum of the four scores is 5 .
OCR S1 2012 January Q3
3 A random variable \(X\) has the distribution \(\mathrm { B } ( 13,0.12 )\).
  1. Find \(\mathrm { P } ( X < 2 )\). Two independent values of \(X\) are found.
  2. Find the probability that exactly one of these values is equal to 2 .
OCR S1 2011 June Q3
3
  1. A random variable, \(X\), has the distribution \(\mathrm { B } ( 12,0.85 )\). Find
    (a) \(\mathrm { P } ( X > 10 )\),
    (b) \(\mathrm { P } ( X = 10 )\),
    (c) \(\operatorname { Var } ( X )\).
  2. A random variable, \(Y\), has the distribution \(\mathrm { B } \left( 2 , \frac { 1 } { 4 } \right)\). Two independent values of \(Y\) are found. Find the probability that the sum of these two values is 1 .
OCR S1 2012 June Q8
8
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 30,0.6 )\). Find \(\mathrm { P } ( X \geqslant 16 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 4,0.7 )\).
    (a) Find \(\mathrm { P } ( Y = 2 )\).
    (b) Three values of \(Y\) are chosen at random. Find the probability that their total is 10 .