If you toss a fair coin six times, what is the probability of get...

Question

If you toss a fair coin six times, what is the probability of getting all heads?

Accepted Answer

Hey everyone in this problem. We have a card that has two fair sides Gray and peach. And were asked to calculate the probability of getting all peaches if you were to flick the card nine times. Alright. So when we have a problem like this where we have two outcomes. We can call one a success and one a failure kind of thing. Um We want to go ahead and use the binomial process. Okay. This is a binomial situation and in a binomial situation the probability of something happening is going to be equal to and choose K. Okay, We're N is the number of trials and K Is the number of successes A P The probability of success to the exponent K Times 1 -2. Which will be the probability of failure to the N minus K. Okay. And so this is going to give us the probability of getting K successes in n trials. Alright. So how does this translate to our problem? Well, okay. And it's going to be the number of trials. Okay. Now we're told that we're going to flick the card nine times. And so in our problem we're gonna have N is equal to nine. Hey, what about K. K? Is the number of successes? Okay, So we're gonna consider that a success is getting peach. Okay. We want to get all peach. We're gonna consider the success to be getting peach. So if we want to get peach every single time then K is going to be nine as well. Okay, we're gonna do this nine times. We want nine peaches. All right. And we have the probability of success. Okay. We're told that this is a fair card. What that means is that the probability of getting gray with the probability of getting peach is the same. Okay. It's a 5050 shot. And so our probability going to be 0.5. Okay, key is nine one minus P. F p is 10.51 minus P. Is also 0.5 K. So this is the probability of failure, which in this case would be the probability of getting gray To the end of -K. They are both nine. So we get to the zero. Okay, so let's work this out. nine Choose 9. So this is essentially we have nine things. How many different ways can we choose? nine things from? Our nine things. We can only choose one way. Right? We have to choose all nine. Every time has to be a peach. And so this is just gonna be one. Okay, 0.5 to the exponent nine. Well, we can write 0.5 as 1/2 To the exponent nine and 0.5 to the zero is going to be one. Okay, so we just have one half to the exponent nine. One to the exponent nine is just going to be one in the numerator and two to the exponent nine is 512 and the denominator. And so we get the probability of getting all peaches from our nine card flicks is one over 512 so that's going to be answer. B. I hope this video helped. Thanks everyone for watching. See you in the next one.