Find parameter from Poisson probabilities

A question is this type if and only if it asks to find the parameter λ of a Poisson distribution given relationships between probabilities (e.g., P(Y=a) = kP(Y=b)).

2 questions

CAIE S2 2019 November Q5
5
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
  3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
    (a) Write down an inequality in \(n\).
    (b) Hence or otherwise find the largest possible value of \(n\).
Edexcel S2 2016 June Q2
2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)
    1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
    2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
  2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
  3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)