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.

$P(X=1)$ is simply the probability of Bob reaching level 1, which is 1 since he starts at level 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

$P(X=j)=p_1{p}_2\cdots p_{j-1}(1-p_j)$.

Since the game has only $7$ levels,

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