Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Probability computing Empirical probability. The empirical probability of an event E donated by P of E is the observed number of times E occurs divided by the total number of observed occurrences or how frequently an event occurs. Computing theoretical probability, P of E is the number of outcomes in event E divided by the number of outcomes in the entire sample space S the probability of an event not occurring P naughty E is equal to 1 -, P of E1 being 100% of the possible outcomes. Probability or probabilities with mutually exclusive events. If A&B are mutually exclusive events, then probability of A or B is just equaling probability of A plus probability of B. You we can use set notation here. So PA union B is PA plus PB probabilities with events that are not mutually exclusive. If A&B are not mutually exclusive events, then P of A or B is P of A + P of B -, P of A&B. We also can use set notation here probabilities with independent events. If A&B are independent events, then P of A&B is P of A * P of B. Let's look at some examples. If we are given a graph of the marital status of the United States population ages 18 or older in millions. If we wanted to find it populate or the probability of a person is divorced, we're going to look at the total of the divorce 21.7. That's how many males and females over the total, so 21.7 / 22.5 or about 10% probability that a person is divorced. If we wanted a widowed male, we're going to look at the widowed male 2.7 over the total again, so 2.7 / 212.5 or approximately .01. So if we chose a person randomly, we would have the probability that they were a widowed male being about 1% or .01. How about among those who are divorced, the probability of selecting a woman? So now we're going to look at divorced women versus the total divorce. So 12.7 / 21.7 or approximately .59 or 59%. If we took the group of divorced people and we just randomly chose a divorced person, it'd be 59% likely that it would be a female. A die is rolled. What's the probability of getting A2? Well, twos occur one out of six times, so the probability of getting A2 is 1/6. We could also think of this as the probability of not getting a 2 taken away from 1:00. So if we had one minus the probability of not getting A2, there are five other numbers that occur, or 1 -, 5 six, hence 16. What about the probability of getting a number greater than 4? Well, we might get a 5 or we might get a six, and the probability of getting A5 is 1/6 and the probability of getting A6 is 1/6. So 1/6 plus 1626 or one third. We could also do this by the probability of not getting. If we had one, we could not get a 5 or 6. So we'd have, if we had a one out of six or A2 on the roll, or a three on the roll or a four on the roll, we would have 1 -, 4 six, which would also be 1/3. How about the probability of a number less than zero? Well, on the die we have to have one through 6. So that's going to actually be a 0 probability 2 numbers whose sum is 4. So looking at the possibilities, we can see that 1322 and three one are the only possibilities that give us a sum of 4. So 3 out of the total possibilities of 36 or 112th. How about two numbers whose sum is 7? That's going to be this diagonal here. 123456 6 out of 36, or one sixth probability that your sum is 7. Two numbers whose sum is one. We can't get a sum of 1. The smallest we could get is 2, so 0 / 36 or a probability of 0. A poker hand consists of five cards. Find the total number of possible 5 card poker hands. So if we have 52 cards in our deck and we want to choose five of them, we'd have 52 factorial over 52 -. 5 factorial times 5 factorial, or 2,598,960. So that's the total number of possible 5 card poker hands. What if we wanted a diamond flush? A diamond flush is A5 card hand consisting of all diamonds. So now instead of 52 cards, we're going to choose amount of 13 cards. So we want all 5 cards to be diamonds. SO13C5 or 13 factorial over 13 -. 5 factorial times 5 factorial 1287. Find the probability of being dealt a diamond flush. Well the diamond flush is going to be the probability of getting a diamond flush out of the possible poker hands total. So it's about .0005 or .05% standard deck of cards. What's the probability of not drawing an ace? Well, one hole minus there are 4 aces in a deck, so 1 hole -4 out of the 52 or 1 minus 113th. So the probability of drawing something, probability of not drawing an ace is 12 thirteenths. How about probability of not drawing a picture card? Well, how many picture cards are in a deck? We have the Jack, the Queen, and the King, so there's three in each suit. So there's 12 picture cards out of a total of 52. So 1 -, 12 / 52, which is going to give us 1 -, 3 thirteenths or 10 thirteenths. We could also think of these if we wanted to think about how many cards are not picture cards. So if there are 13 in a suit and three of them are picture cards, we know that 10 thirteenths would be how many are left? How about brought probability of a red card or A7? Well, there are 26 red cards. There's diamonds and hearts. There's also 4 sevens in a deck. Now two of those sevens happen to read red cards, so we need to subtract the two that are repeats. So here we're going to get 2650 twos plus 4 -, 2 and that's going to give us 2850 twos. And if we reduce that, we're going to get 7 thirteenths. Thank you and have a wonderful day.