Sum of three or more Poissons

1 A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10 -second period have the independent distributions \(\operatorname { Po } ( 0.4 ) , \operatorname { Po } ( 0.1 ) , \operatorname { Po } ( 0.2 )\) and \(\mathrm { Po } ( 0.5 )\). Find the probability that the total number of passengers arriving at the four entrances to the bus station during a randomly chosen 1 -minute period in the morning is more than 3 .

4 The number of lions seen per day during a standard safari has the distribution \(\operatorname { Po } ( 0.8 )\). The number of lions seen per day during an off-road safari has the distribution \(\operatorname { Po } ( 2.7 )\). The two distributions are independent.

1. Susan goes on a standard safari for one day. Find the probability that she sees at least 2 lions.
2. Deena goes on a standard safari for 3 days and then on an off-road safari for 2 days. Find the probability that she sees a total of fewer than 5 lions.
3. Khaled goes on a standard safari for \(n\) days, where \(n\) is an integer. He wants to ensure that his chance of not seeing any lions is less than \(10 \%\). Find the smallest possible value of \(n\).

1 The numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.

1 The numbers of \(\alpha\)-particles emitted per minute from two types of source, \(A\) and \(B\), have the distributions \(\operatorname { Po } ( 1.5 )\) and \(\operatorname { Po } ( 2 )\) respectively. The total number of \(\alpha\)-particles emitted over a period of 2 minutes from three sources of type \(A\) and two sources of type \(B\), all of which are independent, is denoted by \(X\). Calculate \(\mathrm { P } ( X = 27 )\).