2. Mr. Curry is playing basketball. Suppose that, on the average, he makes two successful three-point shots out of every five attempts. What is the probability that he will be able to make at least two successful 3-point shots in a row out of four attempts?
Answer: $\frac{44}{125}$
Solution: There are only a few possibilities to the contrary. Note that if Mr. Curry shoots at least three, for certain, two of them will be consecutive. Hence, we need consider only the cases wherein he makes two shots or less.
No shots made: The probability of this happening is simply $\left(\frac{3}{5}\right)^{4}=\frac{81}{625}$.
One shot made: This one shot can be any one of the four shots, and so the probability in this case is $4 \cdot \frac{2}{5} \cdot\left(\frac{3}{5}\right)^{3}=\frac{216}{625}$.
Two shots made: There are $\binom{4}{2}=6$ ways for Mr. Curry to pick two out of four shots to make, and three of those have the two shots next to each other. Hence, the probability Steph makes two shots, not consecutively, is $3 \cdot\left(\frac{3}{5}\right)^{2}\left(\frac{2}{5}\right)^{2}=\frac{108}{625}$.
Adding all of these gives Mr. Curry a $\frac{81+216+108}{625}=\frac{81}{125}$ chance of not making two consecutive shots. Hence, the chance he makes two consecutive shots is $1-\frac{81}{125}=\frac{44}{125}$.