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Question

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Answers

A special deck of 16 cards has 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to four. A single card is drawn at random. Find the following probabilities. a. The probability that the card drawn is a two or a four. b. The probability that the card is a two or a four, given that it is not a one. c. The probability that the card is a two or a four, given that it is either a two or a three. d. The probability that the card is a two or a four, given that it is red or green.

All right, so in this problem we are looking again at um Probability of certain events happening in this event, we have 16 cards that have four colors blue, yellow, green and red. And they're all numbered 1-4 apiece. So kind of like you know deck is what I'm sort of thinking. So the first event we're looking at is B which is when the card is blue and therefore outcomes in which we would get a blue card, we could get a blue one, the blue too, the blue three Are the blue four for the red card, We could kind of along the same vein. Red one, red too, red three, our red four. Now the next one is a one or two, there's gonna be more outcomes. Um and that means we could have a blue one or the blue too, The Red one or the red too The Green one or the green too Yellow one or the yellow to that one says that most too. So that would be either a one or two of any of the colors. So for part B we start looking at some of the intersections, compliments and unions of these things. So the first one is the intersection of BNR so that means that we have a card that is blue and red and that is um not possible. Okay so there is no outcomes of that. So that would be the empty set there. The next one is um blue or red, that would be the blue one, the blue too, the blue three Blue four or the Red one. Red too. Red three Red Force. And there's a big difference there when we go from intersection to union because it starts to include a whole lot more. Um Next we have the intersection of B. And N. Which means that it's blue and most too that is just the blue one or the blue too. And next we have our union in which would be the red one and the red to kind of along the exact same way the complement of B. Which means not blue anything. Not blue. That would be the red one, the red to all the way to read for The Green one, the green too green three Green four. And then same for yellow. It's a yellow 1-4. So everything but the blue cards And then the last one is the compliments of be union or which is anything but these so that would be any of the yellow or the green. So that would be green, one green to green, three. Green four or Yellow one. Yellow too Yellow three yellow for party. We're talking about the probabilities of these. I'm actually just going to write the probabilities out of here. So the first one there was no way that could happen. So that would be zero out of 16. The next one There are 123-45678 outcomes in that event. So that's eight out of 16 which is 1/2 Next to the next two, we have two out of 16, which both reduced to 1/8. Mhm. Um here we have everything but blue. That's going to be 12 out of 16, which is 3/4 And then the next um the last one we have is eight possible outcomes out of 16, which is one half, so half the colors, right? We're not taking blue or read. That means we are taking green and yellow, which is half of those. So that was part C. And then part D determine whether B and N are mutually exclusive.

Mhm. Okay. So for this problem um we want to find the probability of drawing a red card from a standard deck of 52 cards. Okay, so ah red cards are the hearts and diamonds And there's 13 of each. So 26 red cards total. Okay. And then probability probability means the total of successes over total of everything. Total success over overall over the total. So success is is 26, 26 Red cards over the total number of cards 52. This reduces down to 1/2. Okay. And then finally, to put the answer in probability notation, that's simply you write a big capital P. The probability of a red card is one half or 50 mm.

One card is selected at random from an ordinary deck of 52. We're going to let a stand for the event. A face card is selected. We're going toe. Let be stand for in the event that a king is selected and we're going toe. Let's see, be the event that a heart is selected. Okay? And with this information, we're going to determine various conditional probabilities. So let's start with part a part. A. Is the probability of event Be so keep in mind there are 52 cards, and probability is always going to be your favorable over your possible. So in this case, there are 52 possible cards, and the favorable is that we select a king and there are four kings in the deck. So the probability of event be would be four out of 52 which simplifies toe 1 13 and as a decimal, that is approximately equal 2.77 part B. The probability of event be given that we know a okay, so we know that we're dealing with a face card. So the face cards are the jacks, the queens and the kings of each suit. So we know there are 12 possible cards because there are 12 face cards. Of those 12 face cards, four of them are kings. We have the king of diamonds, the king of hearts, king of spades and the king of clubs, which simplifies down into one third or approximately equal toe a probability of 0.3 three three. Well, let's go to Part C. What's the probability of be given that we know? See? So this time we know that card is a heart. Well, there happened to be 13 hearts in the deck, and we want a king as the favorable Well, there's only one king of hearts, so therefore, the probability of be given see would be 1/13 which is approximately 0.77 Letter D. What's the probability of be given that not a so not a means? It's not a face card, so not a means that there are 12 less cards in the deck, so there are 40 cards that air not face cards. If they're not face cards, then we'll never get a king. So therefore, the probability of be not a is going to be zero letter E. What is the probability of a well again probability is favorable over possible. There are 52 cards in the deck, and there are 12 face cards as are favorable, and that will reduce down to three out of 13, which, as a decimal, is approximately 0.231 Let her act. What's the probability of a given B? So be Waas that it was a king. So there are four possible kings in the deck. So when we're drawing from those, what's the chances of them having or being a face card? Well, all four would be face cards, so the probability of a given B would be one part G. What's the probability of a given that we know? See, we'll see was a heart, and we know that there were 13 possible hearts of those 13 possible hearts. How many are face cards? Well, there's the jack of hearts, the queen of Hearts and the king of Hearts. So are favorable. Is three making our probability three out of 13, which is approximately point to 31 and the final part to this problem, Part H. What is the probability of a not be well, not be means it's not a face card. Sorry, not being means it's not a king, so that removes four cards from the deck. So we now have 48 possible cards. And if we've removed the four Kings from the deck, we have removed four of the face cards. So there's only eight possible fakes cards leftover that we could access, as are favorable cards, so that simplifies down into 16 which is approximately 0.1 six seven.

So in this question, we are given a deck of cards, 16 cards for our blue, for yellow, for green and for red, And the four cards are numbered one through. Mhm. So we have the four types of cards numbered one through four. And first we're asked to find the probability that the card drawn is ready. So we have 16 cars. And the probability that the card drawn is red is basically for over 16 4.25. Now for part we were asked for the probability that the card is red, given that it is not green. So we know a condition. So the condition is that the card is not green. So that basically leaves us with 12 cards remaining and the property that the card is red given that it's not green. So that's basically four out of 12, 1 3rd or 0.33 now in part C, whereas that the probability that the car is red given it is neither read nor yellow, so neither read nor yellow leaves us with four blue and four green cards. So the product is that the card is red given that is neither here nor yellow is basically zero. And in party were asked the probability that the card is red given that it is not a four. So we basically have four cards that are four, and we're told that it's not a force who were left with 12 cards holding that it's red. It basically leaves us with three remaining red cars. So that gives us a probability of 0.25 So we have the answers to a e C and D.


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