All correct here in this problem, we were flipping some points and we ‘d like to find some probabilities here from this probability distribution serve. Yeah. First, let ‘s list out the sample distribution space. You flipped three coins. You could have three heads. The first base two could be heads in the final one tales. First one had second gear tales. Third one head, tails. Heads, Heads. Okay. You could have heads, tails, tails, tails, heads, tails, tails, tails, heads, entails, tails, tails. So those are all of our possible different outcomes here. And we ‘re told that we made a dollar. If the first ones heads, I ‘ll put these in blue here. So we might a dollar with all those outcomes. We made $ 2. If the first heads is the second one indeed that those who make $ 2 we make $ 3 if the first heads is the third base one and then we do n’t make anything if they ‘re all tales and actually we lose the dollars, we do n’t need to put negative $ 1. I do n’t know. last. So get ‘s make our distribution from this now. Why one is the total of heads ? We could have 012 or three heads. My two is the money that we ‘ve earned. And so we could lose a dollar, Make it out $ 2 right ? $ 3 now beginning lease ‘s start with. If yttrium one is zero. If why one is zero, that means there were no heads. So it would only be three tails. And the lone option is we lose a dollar. And thus one out of eight times we lost the dollar with zero heads. immediately that is the only way to end up with no heads. And thus we have zeros everywhere else here. And that ‘s besides the only way to lose the dollar. So we have zeros elsewhere there. now I suppose that we had one hit, We had one head here here and here note that each of those have different outcomes. One of them, we want a dollar, one of those. We want to know one of them. We want $ 2 and one of them. We want $ 3. now when there were two heads repository hera, here and here, two of those times we want a dollar, that ‘s two out of eight here. One of those times we won $ 2. That ‘s one of me. none of those times. We won $ 3. Mhm. And then let ‘s see what happens. Whenever we had three heads, we had three heads, we won $ 1. It ‘s one out of eight times one that and we never 12 or $ 3 when we had three hits. And so this is part a This is your probability distribution function here. This is what it should look. Mhm. Long gripe. We want to find the probability that fewer than three heads will occur and we went a dollar or less, so fewer than three heads and we went a dollar or less. What that means ? We need to be watching here. Okay, that is everywhere that we want a dollar or less. And there are 01 or two heads. therefore there is 18 18 in two ways. Those are all the probabilities that are there in that chart. 1818 and two ways adding those all together gives us one half. That ‘s our probability. here is one.