# Two boys pick a subset of $40$ toys that they like. They can pick the same ones. What is the probability that they picked three same toys or more?

Two boys pick a subset of $$40$$ toys that they like. They can pick the same ones. What is the probability that they picked three same toys or more? My answer would be $$\frac{ \sum_{ i =3}^{40} \binom{40}{i} 3^{40-i}} { 2^{40} 2^{40}}.$$ Is that right? I would first pick the same toys that they picked, then for each of the remaining toys, I would either give to the first boy, second boy or nobody.

• You are correct, but it takes less computation to compute the probability that they do not pick three or more of the same toys--i.e., the probability that they pick zero, one, or two toys in common. – awkward Apr 15 at 12:17

You can also rewrite the formula to be more clear as: $$$$\sum_{i=3}^{40} {40 \choose i } \bigg(\frac{1}{4}\bigg)^{i}\bigg(\frac{3}{4}\bigg)^{40-i}$$$$
As each toy has a probability $$1/4$$ to be chosen by both boys, and $$3/4$$ to be chosen at most by 1 boy and $$i$$ toys can be chosen in $${40 \choose i }$$ ways.