Pedro randomly picks a perfect square between zero and 700. The table shows the probability that the number lies in different ranges. Part a) Find the value of 𝑝.
The clue here is the fact that all of the perfect squares — remember that’s just a square number that’s a result squaring integer — lie between zero and 700. This means that all possible outcomes are included in this table. Since these events are mutually exclusive — that is to say, you can’t choose a number that lies in two ranges — in turn that means that the sum of all these probabilities is one.
We can find the value of 𝑝 by forming an equation to represent this information. We can say that 0.32 plus 0.20 plus 0.12 plus 𝑝 plus 0.12 plus 0.08 plus 0.08 is equal to one. And we can simplify this by adding the numbers together. A column method might be ideal here.
Remember when we’re adding decimals, we must make sure that the decimals themselves line up. We can add a decimal in the answer directly under the decimals in the question. Two plus two plus two plus eight plus eight is 22. We added a two in the hundredths column and carried a two. Three plus two plus one plus one plus the extra two is nine. And all these zeros just give us zero. This means our equation becomes 0.92 plus 𝑝 is equal to one.
To solve for 𝑝, we’re going to do the opposite of adding 0.92. We’re going to subtract 0.92 from both sides of this equation. 0.92 minus 0.92 is zero. So 𝑝 is equal to one minus 0.92. Now, this is probably a problem we can work out in our heads. But a column method can be a nice way to work it out again.
This time we’re going to do 1.00 minus 0.92. It’s always sensible to add the zeros after the decimal point to make sure the numbers are the same length. We can’t do zero minus two. So we have to borrow from the first number to the left which isn’t zero; that’s a one. So the one becomes zero. And we add a one to the second zero.
We’re now going to borrow from 10. 10 take away one means this becomes nine. And then, we add the one to the first zero. 10 take away two is eight. Nine take away nine is zero. We put a decimal point here. And zero take away zero is zero. This means one minus 0.92 is 0.08. And the value of 𝑝 in our table is 0.08.
b) Calculate the probability that the perfect square is less than 300.
For the perfect square to be less than 300, it could be between zero and 100, 100 and 200, or 200 and 300. In fact, this strict inequality tells us that the numbers in this range are greater than or equal to 200, but they are strictly less than 300.
Here, we can use this rule: when two events are mutually exclusive, remember we said at the start that just means they can’t happen at the same time. We find the probability that one event or the other occurs by adding their respective probabilities.
To find the probability the number lies in the range zero to 100 or 100 to 200 or 200 to 300, we’re going to add their probabilities. That’s 0.32 plus 0.20 plus 0.12. Let’s use a column method again. Two add two is four. Three plus two plus one is six. We add the decimal point. And all these zeros give us zero. This means that 0.32 plus 0.20 plus 0.12 is 0.64.
And the probability that the perfect square is less than 300 is 0.64.