Geometric distribution probability

Find probabilities involving the number of trials until first success, using geometric distribution (e.g. repeated coin tosses until tails).

2 questions

CAIE S1 2020 June Q5
5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.
  1. Find the expected value of \(X\).
  2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
  3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
    On a different occasion, a pair of fair coins is thrown 80 times.
  4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.
CAIE S1 2020 Specimen Q5
4 marks
5 A fair six sid dl e,w itlf aces mark dress s ther im imes.
  1. Use ara \(p\) in matin of id b pb b lity b ta 3 s ob ain of ewer th rㅇs imes. [4]
  2. Js tifys se 6 th ap ox matin pe rt (a). Ora \(\mathbf { h }\) b roccasity he same \(\dot { \mathbf { d } }\) e is th \(\boldsymbol { w }\) ep ated y il a \(\mathbf { 3 } \mathrm { sb }\) aie d
  3. Fid b pb b lity b tb ain g ʒ eq res fewer th \(n\) st \(h\) s.