Q.4.61 - A First Course in Probability

Question

The probability of being dealt a full house in a hand of poker is approximately $.0014$. Find an approximation for the probability that in $1000$ hands of poker, you will be dealt at least $2$ full houses. 

Step-by-Step Solution

Answer

An approximation for the probability that in $1000$ hands of poker, we will be dealt at least $2$ full houses is $0.408$.

Step 1: Given information

We are given that we have $n=1000$ autonomous Hands of poker, wherein every one of them we have $p=0.0014$ probability to get a Whole House.

Step 2: Calculation

So on intermediate, we will hold $\lambda =np=1.4$ Full Houses in this session. Hence, we can use the Poisson approximation to get the demand.

So, let $Y~Pois\left(\lambda \right)$.

We have that

$P(Y\ge 2)=1-P(Y=0)-P(Y=1)=1-e^{-\lambda }-\lambda e^{-\lambda }\approx 0.408$

Step 3: Final answer

So there is approximately $0.408$ probability to get at least two full houses in $1000$ hands.