I have encountered several question of the following format. I have no trouble answering the first half but second half I have no clue on how to proceed.
a: If you roll 5 standard six-sided dice, what is the probability of getting at least three 2s?
b: If the roller gives you $10 on getting at least three 2's but asks only $2 in return if you lose. Would you take the bet? Why or why not?
First part is just an application of binomial theorem and you are done. The probability is 0.03549. But how to go about second part. Does it make sense to also take into consideration that if I win only once and lose other 5 times then also I will have not lost anything in the end. It will be evened out.
Answers with complete explanation would be the best. If not, then please provide pointers to relevant readings.
In order to answer the second part, you need to make sure you understand the concept of "expected value". The implied assumption of part b) is that you're playing the game over and over and want to see what's the average profit/loss you make over all these rounds.
With this in mind, the expected value of a bet with two outcomes is just $$ \begin{align} E &= (\text{Win Amt.}) \times P(\text{Win}) - (\text{Loss Amt.}) \times P(\text{Loss}) \\ &= 10(.03549) - 2 (.96451) \\ &= .3549 - 1.92902 \\ &\approx -1.6 \end{align} $$
Hence on average, you lose about $1.6 for every round of the game. Should you play?