Video: CBSE Class X • Pack 3 • 2016 • Question 29

CBSE Class X • Pack 3 • 2016 • Question 29

03:33

Video Transcript

A number 𝑥 is selected at random from the numbers one, two, three, and four. Another number 𝑦 is selected at random from the numbers one, four, nine, and 16. Find the probability that the product of 𝑥 and 𝑦 is less than 16.

Okay, so there are four possible outcomes for the value of 𝑥 picked. One, two, three, and four are all equally likely to be selected. But independently of randomly selecting a value of 𝑥, a random value of 𝑦 is selected from the numbers one, four, nine, and 16. Each of these values of 𝑦 is equally likely to occur..

There are four possible values of 𝑥 and four possible values of 𝑦. And therefore, there are four times four, which is 16 possible pairs of values of 𝑥 and 𝑦. Our task is to find the probability that the product of 𝑥 and 𝑦 is less than 16.

When dealing with equally likely outcomes, the probability of an event is a fraction, where the denominator is the total number of equally likely outcomes and the numerator is the number of those equally likely outcomes which are favorable; that is, which are part of the event.

In our case, the event that we’re interested in is the event that the product of 𝑥 and 𝑦 is less than 16. What is the total number of equally likely outcomes? Well, we’ve seen it’s 16. There are 16 equally likely pairs of 𝑥 and 𝑦.

How many of those pairs are favorable; that is, for how many of those pairs is the product of 𝑥 and 𝑦 less than 16? Well, let’s count them. For each pair of 𝑥 and 𝑦, we find their product. The product of one and one is one, the product of two and one is two, the product of three and one is three, and the product of four and one is four.

We continue with the next row. The product of one and four is four; of two and four is eight; of three and four is 12; and of four and four is 16. One and nine is nine. Two nines are 18. Three nines are 27. And four nines are 36. And finally, one 16 is 16, two are 32, three are 48, and four are 64. So these are the products you get from the 16 equally likely pairs of 𝑥 and 𝑦.

Now, how many of them are less than 16? Let’s count them. One is less than 16 as are two, three, four, four, eight, and 12. However, 16 itself is not less than 16. So we don’t count that. Nine, however, is less than 16. But the rest of the products are all greater than or equal to 16 and so are not counted.

So how many favorable outcomes do we have? Well, we have eight. There are eight equally likely pairs of 𝑥 and 𝑦 for which their products 𝑥 times 𝑦 is less than 16. All this left to do is to simplify this fraction. Eight over 16 is one-half.

So when a number 𝑥 is selected at random from the numbers one, two, three, and four and another number 𝑦 is selected at random from the numbers one, four, nine, and 16, the probability that the product of 𝑥 and 𝑦 is less than 16 is a half.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.