If there is a lottery every week and the chances of winning are $0.07$. Daniel decided to buy a lottery ticket every week till he will win or till he will buy $6$ tickets. $X$ is the number of weeks in which Daniel bought a lottery ticket. What are the chances that $X$ is an even number?
I am not sure how to start. I thought to do $0.07$ by the power of $1$ and then by the power of $2$ etc to check the chances to win in every week, but I'm not sure how to check if a week is an even number and to win the lottery. Is it some sort of Bernoulli experiment?
Best Answer
The number of days here are limited, so you can calculate the probability for each day extra and then sum them up.
You can split the probability to three cases: losing the 1st day and winning the 2nd day, losing 1st-3rd day and winning the 4th day, loosing 1st-5th day. All these cases are disjoint. Let p be the chance to win ($p=0.07$). What you get is: $$ (1-p)p + (1-p)^3p + (1-p)^5 = 0.93\cdot0.07 + 0.93^3\cdot0.07+0.93^5 = 0,817093359 $$