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A man fires two shots with a rifle at a target.The probabilitythat he will hit the target with the first shot is 0.3,with thesecond shot 0.5 and with both shots 0.2...

Question

A man fires two shots with a rifle at a target.The probabilitythat he will hit the target with the first shot is 0.3,with thesecond shot 0.5 and with both shots 0.25.If he hit the target withthe first shot what is the probability that he will hit it with thesecond as well

A man fires two shots with a rifle at a target.The probability that he will hit the target with the first shot is 0.3,with the second shot 0.5 and with both shots 0.25.If he hit the target with the first shot what is the probability that he will hit it with the second as well



Answers

An athlete is shooting arrows at a target. She has a record of hitting the centre $30 \%$ of the time. Find the probability that she hits the centre a) with her second shot b) exactly once with her first three shots c) at least once with her first three shots.

So in the given question, we are told that the probability that a shooter heads a shooter hits the target hits target is given as three by five. And we are told to find the probability that he hits five targets five targets when he shoots, when he shoots 10 times. So what is the probability probability that he hits five times five? Not five targets five times when he shoots the target 10 times. Right. Sure. This is the probability that we need to find. So what we have over here is a binomial trial, right? And what a binomial trial is that he said trial, in which there is only two outcomes, only two outcomes, which means it is either a success odd of failure. Right? So the type of trying in which there are only two outcomes. And if one outcomes with one of the outcomes, we can say, we can categorize it as a success and the other one we can say as a failure. So if over here, the shooter hits the target, we can say it is success. And if he misses the target, we can say it is a failure. Right? So in such a trial, if we have, if we have in trials that are being followed and we have X number of successes. The property that there is X number of successes can be calculated by taking the property that there is X. Number of successes can be calculated by taking n C x times Q raised to end minus X times P raised two X over here. And is the number of trials and Q. S the probability of failure and ps the probability of success. So this is the data that we have and now we can use this method, use this trial, they use this formula to find the probability over here. Right? So what we have over here is we can take any equal to 10 and the success probably tp as three by five. The failure would then be one minus speed which is equal to two by five. Sure. And now we can right P of X is then equal to Yeah. The effects we have. We have to have five successes. Right? So the property of having five success Having five success is equal to N C. Five. That is 10 C. Five times Q. Raised to and minus cure raised two and minus X. Which is to buy five, raised to 10 minus five times be raised to X. Which is three by five raised 25 Right? So this is equal to 10 C. Five times two by five raised to five times three by five raised to five. So this is the required probability, right? So we can simplify this by taking by writing at us 10 c five times six raised to five, divided by five times five raised to 10. So this is what we have as the required probability, right? And in the question we have four options out of which the option number A uses the probability asked 10 c five mm hmm. 10 C five times six raised to five, divided by five, raised to 10. So the required answer is required option, The correct option? This option A And I hope you understood the method. Thank you.

Okay, so we're looking for the probability that this archer who is this started 80% of the time. So P equals 0.8 and he shoots 70 years. So we have unequal seven. And first off, we want where he never hits the target. That's our equal. Zero soapy are equal zero, which will be 7 to 0. Have our 0.8 is here with her 0.2 to the seventh Power. Then once we find what that probability is going to be, No, but your answer. Which I'll go ahead and write up the end, but we'll go on and we're talking about probability of the so Probably be that he had to target each time. So he hits it every time. So we'll do p of Article seven. Also, this is gonna be equal to one because we'll have seven factorial over seven factorial, and we'll have a very similar case here because we have seven factorial over seven factorial you a factorial. So this will be one and then we'll have 0.8 to the seventh time, 0.2 to the zero hour and then for apart. See, it says hits the target more than once. So then I'm going to propose that the easiest way of doing this one so probability are greater than one would be just one minus p. R. Equals zero minus p of our equals, one which we already which will already have p of our equals. One. What's so? I guess I finally have to actually go ahead and figure out what that number is in that. Okay, so that number is going to be 0.0 or 1 to 8. And also, I went ahead and found a probability for our equal seven, which will be 0.2097152 And now we want to find the probability of articles one for this problem. So our probability of articles one is going to be this seven factorial over one factorial over six factorial. And then we'll have our 0.8 to the one power time, 0.2 to the sixth power. But that's gonna end up giving us is going to be seven times 70.8 times 0.2 to the sixth. I'm getting to be about zero point 000 3584 And so when we do one minus that 0.3584 minus the 0.1 to 8. We'll end up getting a very close number to just one. So this is our answer right here. Actually, I'll go ahead and box in, just so it's very clear which ones we have, then finally it for our last one. He hits the target at least five times. I'm gonna have to open another white board for this one. So far, a D. We have p of our it's the target at least five times. So then that's saying that we're gonna have greater than or equal to five once again. Just as a reminder, I'll put all the way over here. We're shooting this seven times and we are shooting this with a probability of 0.8. So what we're gonna do here is we want the probability are you was five plus, the probability are equal six, plus the probability of R equals seven, which we already know, what probably of our equal seven s. So we'll just have to find what probably are equals five in the probability of article Sixes. So let's go. Heading in these set up real quick. Seven factorial over too. Yeah, Factorial. Then we'll have our 0.8 to the fear zero point to square, Then over here will have seven factorial. Then we'll have six factorial one factorial your point A to the 6th 0 point to the one And then once we go ahead and figure out what this gonna be, so will do. Seven Choose five times 0.8 to the fifth turning 0.2 square. It's getting to be about zero point 27 5 to 5 12 Then over here I'm getting seven times 70.8 to the sixth times point to which will give me about zero point 367 0016 So we'll add our 0.27525 12. And we will be adding from our first page what we got for P of our equal seven, which is 0.2097152 And what we end up getting for total there. So then, if we have an equal sign here, I'll go ahead and circle it. Just clarity. There won't be bringing it down. Here are total probability is gonna be equal to 0.851968

So in the given question, we are told that there are three people who are firing at a target, right? So there are three people that's uh which they are given us a BNC and there trying to hit a target, right? And it is given in the question that the probability that a hits the target is equal to Is equal to 0.3. And the probability that be hits is equal to 0.4. And the probability that C hits the target is Given us 0.5. And we are told to find the probability that the target is hit right, The property that the target is hit. So we can easily find this probability by taking true the the method of we can take the probability of the property that at least one of them heating. I think the target can be found by taking 1- the probability that none of them had the target, none of them head. And how can we take the probability that none of them had? So we have we are given the question, the probability of each person hitting the target. So we can take P. Of A. And the probability of mm not hitting not hitting. Let's write it over here. Right, So let's write this over here. So the property that may not meeting can be found By taking 1 -1 the probability that A hits. So it is equal to 1 -1 0.3 which is equal to 0.7. Similarly, we can take the probability of B not hitting As 1- probability that be hits the target. So it has given us 0.4. So one minus 0.40 point six and last we have C. So the property of C not hitting can be found by taking one minus the probability that she hits the target. It has given us 0.5 so one minus 0.5 is equal to 0.5. So the probability that none of them is hitting Is the product of 0.7 times 0.6 times zero five. And this this would give us the probability the required probability as 1 -0.7 times 0.6 times 0.5 is 1 -1 0.21. Yeah. And this is equal to zero point 79 And when we look at the question there are four options and of the options, The option that uses the answer as 0.79. This option beak, So option B 0.79 is the correct answer. So I hope you understood the method. Thank you.

All right. So the probability that are Archer never hits the target Out of those seven shots is zero point to the seventh power because we're assuming that each shot is an independent of each other. And so, in order to not hit the target ever, he needs to have missed the target, which is 20% seven times in a row. So and this turns out to be approximately zero point 001% Part B. We're doing the opposite where he hits all seven times. So we do 0.8 to the seventh Power, and that's going to be approximately 20 points. 972%. Part C is asking for the probability that he hits more than once, as in twice, three times, all the way up to seven times. So what we're going to do is find the opposite probable the probability where he hits zero times or one time and then subtract that probability from one. So we're going to do one, minus the probability that he hit zero times. We've already found a part A as 0.2 to the seventh power and the probability that he hits exactly once is going to be seven Choose one times 0.2 to the six power times 0.8 the if does the narrow where he hits exactly once can be hit. Miss Miss, Miss Miss, Miss Miss or Miss Hit Miss Miss, Miss, Miss Miss All are up to Miss Miss Miss Miss Miss Miss Hit So you can see the seven choose one ends up being seven. And if you plug that into a calculator, it will give you approximately 99 point 963%. And now for the final question, which is which is asking for the probability that he hits at least five times. So we're going to do something similar to what we did in part C. Except for the one minus part. So we're going to do 0.8 to the seventh power to take into account the chance that he hit seven times and then seven shoes. Six times 0.8 to the six power times 0.2 plus seven shoes five times 0.8 to the fifth power times 3.2 squared. So when when you plug that in, you're going to get approximately 85. It's 197%


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