A clinical trial for a new vaccine shows a $70\%$ chance of success. Suppose the vaccine is given to three people. Construct the binomial distribution for this situation.

$X$: The number of people in the group

$P(X=0)=0.117$

$P(X=1)=0.193$

$P(X=2)=0.401$

$P(X=3)=0.398$

$X$: The number of successful treatments

$P(X=0)=0.027$

$P(X=1)=0.189$

$P(X=2)=0.441$

$P(X=3)=0.343$

$X$: The total number of treatments performed

$P(X=0)=0.056$

$P(X=1)=0.199$

$P(X=2)=0.584$

$P(X=3)=0.298$

$X$: The number of failed treatments

$P(X=0)=0.135$

$P(X=1)=0.169$

$P(X=2)=0.251$

$P(X=3)=0.343$