For part (a), you might try drawing a
tree diagram. When doing so, consider a simpler case where you pick 2 or 3 people. For the first person, you are given the probability that they're happy, but what is the probability they're not happy? As a hint, the probably of something
not happening is 1 minus the probability of it happening. Put that information into your diagram. If the first person is happy, what is the probability the second one is? What's the probability they're not happy? Where do you think that information should go on your diagram? Place it accordingly. Now, what if the first person wasn't happy? Does that affect the probability of the second person being happy? Why or why not? Does the first person being unhappy affect the probability of the second person being unhappy? Why or why not? Where do you think you'd put this information on your diagram? Now, consider the third person. Put their information into the diagram. Now, what is the probability of the first and the second people being happy? How did you calculate that? How would you calculate the probability of all three people being happy? Are you seeing a pattern? If not, maybe add a fourth person to the diagram and find the probability of all
four being happy.
Now, once you have intuited the pattern (the "formula," if you will) for determining if the first
n people are happy, you can use that to help you calculate the probability of exactly 8 of the 10 people are happy. Note that you'll need to consider multiple cases here. Since 8 people are happy, there are two unhappy people. What if the two unhappy people are #1 and #2? What if they're #1 and #3? How many cases, in total, will you need to consider? Is each of the cases equally likely? Why or why not? How does all of this information help you figure out the overall probability of any 8 of the 10 people being happy?
As for part (b), that's actually probably easier than part (a), although it may not look like it at first. As you know, the probability of something happening is 1 minus the probability of it not happening. Since finding the probability of at least 2 people being happy means finding the probability of 2 people, 3 people, 4 people, etc. being happy, it will certainly be simpler to consider the probability of there being less than 2 happy people. So, what is the probability of 0 people being happy? Using your results from part (a) to help you along, what is the probability of 1 person being happy? What is the probability that less than 2 people are happy? Then, what is the probability of at least two people being happy?
If you get stuck, that's okay. But, when you reply, please include
all of your work, even if you know it's wrong. Thank you.