Video: AF5P3-Q09-309159069283

One day, a village fête holds a game where players throw darts to pop balloons. In round 1, there are 2 balloons at a close distance. Players progress to around 2 if they pop 1 or 2 balloons. In round 2, there is one balloon at a faraway distance. Players win if they pop that 1 balloon. 250 people play the game throughout the day. The frequency tree shows some of the outcomes. (a) complete the frequency tree. (b) Mavis plays the game. Use the frequency tree to estimate the probability that Mavis loses. (c) The organisers have 500 balloons at the start of the village fête. Use the frequency tree to work out how many unpopped balloons are left at the end of the village fête.

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Video Transcript

One day, a village fête holds a game, where players throw darts to pop balloons. In round one, there are two balloons at a close distance. Players progress to around two if they pop one or two balloons. In round two, there is one balloon at a faraway distance. Players win if they pop that one balloon. 250 people play the game throughout the day. The frequency tree shows some of the outcomes. Part a) complete the frequency tree.

There’s also a part b and part c that we’ll come on to. So for part a, what we need to do is complete the frequency tree. I’m gonna start with this space first. And this is the space that shows the number of people who popped two balloons after round one. We can see that coming off this, for round two, is people who have popped one balloon or no balloons. And the values for each of these are 28 and 55, respectively. So therefore, if we want to work out how many people popped two balloons after round one, so the space that we’re looking to fill. Then we’re gonna have to add these two values together because these come off of our space that we’re trying to fill. So we’re gonna have 28 plus 55, which is equal to 83. And we could’ve done that on our calculator.

We could’ve, however, also done it using a written method. So we could have a column method. So 55 add 28. First of all, we add five and eight, which gives 13. So you have three in the unit column and one in the tens column. And that one in the tens column is one that we’ve carried. So now what we do is we add the tens together. So we’ve got five add two plus the one that we carried gives us eight because seven add one is eight. So we’ve got 83. So great. We’ve now worked out how many people popped two balloons after round one. So now, let’s move on to the next space.

Well, the next space we’re looking to calculate is this middle one. And this is the number of people who popped one balloon after round one. Well to work out this value, what we need to do is use a bit of information that we’ve been given. And that information is the first space. Well, it’s also being told to us in the question. And that is that 250 people play the game throughout the day. Now we know how many people pop two balloons because we worked that out. It’s 83. And we know how many people popped no balloons, because that’s 44. Cause it’s already in the tree. So therefore, to calculate the missing value, what we’re gonna do is 250 minus 83 minus 44, which is equal to 123. And we could’ve used our calculator to work this out.

However, we can also double-check it using written methods. So the first thing we’d want to do is add 83 and 44 together. Because then what we can do is subtract the answer of this away from 250. First of all, we have three add four, which is seven. So we put a seven in the units column. And then, we have eight add four, which is 12. So we have one in the hundreds column, two in the tens column. So we get 127. Okay, great. So now, we need to use this and subtract this value away from 250.

So we’ve set up the column subtraction, 250 minus 127. And first what we want to do is zero minus seven. But we can’t do this. So we need to borrow from the tens column. So we’ve got four now in the tens column. And we’ve got 10 in the units column. So we could say 10 minus seven is three. So we put three in the units column. It’s worth noting at this point, be careful. A common mistake if you see a zero and a seven is to think I’ll just do seven take away zero, which is just seven. But you can’t do it that way round. You have to do the top number minus the bottom number. Okay, next, we move on to the tens. We’ve got four minus two, which is just two. And then finally, the hundreds. We’ve got two minus one, which is just one. So therefore, we’ve got 123, which is the number we had previously. So great. We’ve worked out the middle value. We could have also, to double-check, added the three values together. So 83, 123, and 44. And if we do that mentally, we could say that 80 add 120 is 200 and then add 44 is 244. Then add three is 247, add another three would be 250. So yes, it works. And we’ve got the correct value.

Okay, so now let’s move on to our final space. So now, to work out the value of our final space, which is the number of people who popped one balloon in round one and then no balloons in round two, what we need to do is subtract 71 away from 123. And that’s because we know that 71 plus the value we’re looking for should add together to give us 123. So if we do 123 minus 71, we get the answer of 52. So therefore, we’ve completed part a because we’ve found the three missing values and completed the frequency tree. Okay, great. Now let’s move on to part b.

So part b) tells us that Mavis plays the game. And it says use the frequency tree to estimate the probability that Mavis loses.

So first of all, to work out this probability, we need to see how many people lose the game. So there are three areas where players can lose. If players pop no balloons in round one, then they lose. Because we’re told that players progress to round two if they pop one or two balloons. And then, in round two, we’re told that players win if they pop the one balloon. So therefore, they lose if they don’t pop any balloons. So that’s where the three sections come from. So therefore, to work out how many players lost the game, we’re gonna add 55, 52, and 54. When we add these together, we get 151. So therefore, we know that 151 players lost the game.

So now what we want to do is estimate the probability that Mavis loses. So to do that, we’re gonna use the probability formula. And that tells us that the probability of an event, that’s the notation I’ve used here, is equal to the number of desired outcomes. So in this case, the number of people who lose, divided by the total number of possible outcomes. So in our case, it’s the total number of people who played the game. So therefore, we can say that the probability that Mavis is going to lose, and this is an estimate for that probability, is gonna be equal to 151 over 250. And that’s because 151 players lost the game. And 250 players played the game. It’s worth noting at this point that this could’ve also been left as a decimal or a percentage. Okay, great. That’s part b. Now let’s move on to part c.

So for part c), the organisers have 500 balloons at the start of the village fête. Use the frequency tree to work out how many unpopped balloons are left at the end of the village fête.

So what we need to do to solve this problem is first of all work out how many balloons were popped. I’m gonna start with round one. Well, first of all, we can see that, in round one, we had 83 people who popped two balloons. So to work out the number of balloons that were popped, we’re gonna do 83 multiplied by two, which is 166. We can work this out with a calculator. But just to demonstrate how would we’d use a written method to 83 multiplied by two, so we did two multiplied by three, which is six. So we have six in the unit column. And then, two multiplied by eight is 16. So you have one in the hundreds column, six in the tens column. So we get 166. Okay, great.

So next, in round one, we have the people who popped one balloon. So we’ve got 123 of those. And as they popped one balloon each, to work out how many balloons were popped, we just do 123 multiplied by one, which is 123. So therefore, to work out the total number of balloons popped in round one, we’re gonna add 166 and 123, which gives us 289 balloons. Okay, great. That’s round one. Now let’s move on to round two.

It is worth noting at this point that we can move on to round two because there are 44 people who popped no balloons. But we don’t need to work anything out there because, as we said, no balloons were popped. So in round two, of the people who popped two balloons in round one, 28 popped one balloon in round two. So therefore, we can say that 28 multiplied by one balloon that was popped by them, which is 28. And then, once again, we could ignore the people who popped no balloons in round two. So there were 55 of them of the people who popped two balloons in round one.

So therefore, what we’re interested in next is the people who popped one balloon in round one and one balloon in round two. And there were 71 of those. So we have 71 multiplied by one, which equals 71. So 71 balloons were popped. So therefore, the total number of balloons popped in round two is gonna be equal to 28 add 71, which is 99 balloons. Okay, great. So now to work out the total balloons popped overall, we’re gonna add the total balloons popped in round one to the total balloons popped in round two. Well again, we could just use a calculator. But I’m gonna show a written method once more. So we’ve got 289 plus 99. Well, nine add nine is 18. So we put eight in the unit column and carry the one. Well, nine add eight is 17 add the one we carried gives us 18. So again, eight in the tens column this time and one in the hundreds column. And one add two is three. So we get 388 is the total number of balloons popped.

We could’ve also worked that out nice and quickly with a mental method. And that’s because we could’ve done 289 plus 100, which gives us 389, and then subtract one because we only had 99 not 100. Okay great. Have we solved the problem? Well, no. We haven’t solved the problem yet because what we’re looking for is the number of unpopped balloons that are left at the end of the village fête. So therefore, to work out this, what we need to do is take away the number of balloons that were pooped, which is 388, away from the total number of balloons, which was 500. And we’re told that in the question. And when we do 500 take away 388, we get 112. So therefore, we can say that we finished part c, because we’ve answered the question and found out that the number of unpopped balloons left at the end of village fête was 112.

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