Write these probabilities in order, starting with the most likely. The options are two-thirds, 68 percent, seven tenths, or 0.688.
In order to compare our probabilities, what we want to do is make more fractions, decimals, or percentages. Well, the way that I’m gonna solve it is by converting them all to decimals as this is going to be the easiest method. So, first of all, I’m gonna start with two-thirds or two divided by three, which is what two-thirds means. Well, what we’re gonna do is see how many threes are going to two. And I’m gonna use short division. So first of all, threes into two don’t go. So what I’m gonna do is I’m gonna put a decimal point, and I would align these. Then, I’ll put another zero after the 2. Then, we’re gonna see what happens if I carry the two. So now, if I carry the two, I have got 20 and I need to see how many threes are going to 20, which is gonna be six. And there’s gonna be a remainder of two. That’s because three multiplied by six is 18. So we carry out two again. Then, we’ve got threes into 20. And therefore, threes into 20 goes six times again with another remainder of two.
As you can see, this is gonna repeat this pattern. So it’s gonna continue on. So I’m gonna get 0.666. And so, we can also call that 0.6 recurring. But as we’ve got an answer with three decimal places, we’re gonna leave our answer to three decimal places. So we’ve got 0.66. And then, we’d have another six and obviously another six after that. So what we’re gonna around it to? Well, if we look at the third decimal place, we put a line after it. Then it’s the digits to the right of this that’s gonna help us to decide whether we’re gonna round up or round down. Well, if the digit after it is five or above, we’re gonna round up. So in this case, it is a six, so is five or above. So we’re gonna round our six to a seven.
So therefore, we can say that two-thirds is equal to 0.667 and that’s to three decimal places. And that’s what three dp means. Now, we could have also worked this out using our calculator by just typing in two divided by three. And this is the answer we would have got.
For our next answer, we’ve got 68 percent. Well, 68 percent means 68 out of 100 or 168 over 100 because percent means out of 100. Well, if we divide 68 by 100. Again, we can use our calculator. Or if we’re doing it mentally, it means we’re gonna move each five digits two places to the right. Because if you divide by 100, that’s what we do. So therefore, 68 percent is gonna be equal to 0.68. And that’s because if we moved six two place values to the right, it goes from being a ten to a tenths and eight two place values right goes from being a unit to a hundredths. What I’m gonna do is add a zero after this. And the reason we’re gonna do that is because each of our answers we’re gonna want to be with three decimal places. So it’s easier to compare them. You wouldn’t normally do this because it’s just 0.68. But you can add the zero because we know that there’s no more values after this.
So the next one is seven tenths. Well, if we think about seven tenths, again we can use a calculator, this is gonna be 0.7. Also, if we think about what we’re doing, we’re dividing seven by 10. If you divide by 10, you move the digit one place value to the right. So it goes from being a unit to a tenths. So we have 0.7. And again, I can add a couple of zeros because it’s gonna help us when we’re comparing our different decimals. So now, we have all of our values, so all of our probabilities, as decimals. Now let’s compare them.
Before we do, however, we want to see in what order we want to put them. We were told that we want to start with the most likely. Well, if you remember our probability scale, we’ve got zero. That is impossible. Then, we’ve got a half means that you have an equal chance and one is certain. So therefore, greater the decimal, the more likely it is to happen. So what we’re going to do is start with the highest value and work down from there.
So now, if we want to compare our decimals, first we’re gonna look at the tenths column. And we see which one has the highest value in the tenths column. Well, that’s 0.7 or 0.700 because this is the only one with seven in the tenths column. So that’s gonna be our first value. So we can now ignore this row. Now, we move along to the hundredths column. And that’s because all the rest have a six in the tenths column. So now, we can rule out the first row. And that’s because this has a six in the hundredths column, whereas the other two rows have an eight.
So now, finally, we’re looking at the thousandths column. And we can see that eight is greater than zero. So this is gonna be the second largest decimal, which means that 0.680 will be the third largest and 0.667 is going to be the smallest. And again, the reason it’s the smallest is because it had lower than a seven in the tenths column. And then, it had the lowest value in the hundredths column.
Okay, now, we have our order. So let’s write out our probabilities in the correct order, starting with the most likely. Well, first, we have 0.7 or 0.700. Well, this is in fact seven tenths. So that’s gonna be our first probability. Then, next, we have 0.688. And this is just as it is, because that was in a decimal form to start with. Then, next, we have 0.68 or 0.680 which is the same as 68 percent. Then, finally, we’ve got 0.6 recurring or 0.667 if we round to it three decimal places, which is two-thirds.
So therefore, our probabilities in the correct order starting with the most likely are seven tenths, 0.688, 68 percent, and two-thirds.