3.1.6 problem 7

To find the probability mass function (PMF) of $X$, we need to determine the probabilities of Bob reaching each level from 1 to 7.

Level 1 is the highest level reached by Bob if he fails to pass level 1, which happens with probability $1-p_1$. Thus, $P(X=1)=1-p_1$.

For $2\le j\le 6$, $P(X=j)$ is the probability of reaching level $j$ but not reaching level $j+1$. This can be calculated as

$P(X=j)=p_1\cdots p_{j-1}(1-p_j)$.

Bob reaches level 7 if he passes each level from 1 to 6:

$P(X=7)=p_1{p}_2{p}_3{p}_4{p}_5{p}_6$.