Problem: You have two options
1: One successful shot at the hoop
2: 2 of 3 successful shots at the hoop
Let P be the probability of making a successful hit, for what value of P will you choose option 2 over 1.
If P is probability for option 1
Option 2, we can have 2 cases of success, one all 3 shots are successful – P*P*P
or 2 Hits and 1 Miss (HHM, HMH, MHH) 3 cases, each with probability of P*P*(1-P) i.e. 1-P for one miss. as there are 3 cases, total probability is 3*P*P*(1-p). Combining both result for option 2, P*P*P + 3*P*P*(1-P).
To prefer option 2,
P*P*P + 3*P*P*(1-P)>P
Can be solved to quadratic equation
P>1 not possible so
P>1/2 or .5
Choose option 2 when probability of hitting hoop is more than 0.5