Video: Pack 4 • Paper 1 • Question 21

Pack 4 • Paper 1 • Question 21

04:56

Video Transcript

There are 𝑥 counters in a bag. Six of the counters are green. All of the other counters are yellow. Georgia takes two counters from the bag at random. Write an expression, in terms of 𝑥, for the probability that Georgia takes one green and one yellow counter from the bag.

Let’s just read the information in the question again. We’re told that the total number of counters is 𝑥 and that the number of green counters is six. This means that the number of yellow counters is given by 𝑥 minus six. We’re going to use a tree diagram to answer this question as it will help us picture all of the possible outcomes more clearly.

The first counter that Georgia removed from the bag can be either green or yellow and the same is true for the second counter. We need to consider the probabilities for each branch of the tree diagram. On the first branch of the tree diagram, the probability that the counter Georgia takes is green is six over 𝑥 as this is the number of green counters over the total.

As Georgia takes the counters from the bag at random, each counter is equally likely to be chosen. And therefore, we can just count up the number of green and the total in order to find the probability. The probability that the first counter is yellow is 𝑥 minus six over 𝑥 as this is the number of yellow counters over the total. Now, we need to consider the probabilities for the second counter and they’re not the same as the probabilities for the first because one counter has already been removed from the bag.

This means that the denominators for these probabilities are 𝑥 minus one as the total number of counters in the bag at this point has reduced by one. Now, let’s consider the numerator for each fraction.

If the first counter chosen was green, then the number of green counters left in the bag is five. So the probability that the second counter will be green is five over 𝑥 minus one. The number of yellow counters won’t have changed. So the probability that the second counter will be yellow is 𝑥 minus six over 𝑥 minus one if the first counter was green. If however the first counter was yellow, then the probabilities will be different. The number of green counters in the bag won’t have changed. So the probability that the second counter will be green if the first counter was yellow is six over 𝑥 minus one. The number of yellow counters in the bag will have reduced by one. So the probability that the second counter is yellow is 𝑥 minus seven over 𝑥 minus one.

In total, there were four possibilities for the colors of the two counters: green, green; green, yellow; yellow, green; and yellow, yellow. We’re asked to find an expression for the probability that Georgia takes one green and one yellow counter from the bag. But the order doesn’t matter. So it’s the middle two branches of the tree diagram that we’re interested in.

To find the probability of the first counter being green and the second being yellow, we multiply along the branches of the tree diagram, giving six over 𝑥 multiplied by 𝑥 minus six over 𝑥 minus one. We’ll expand the bracket in the numerator, giving six 𝑥 minus 36, but keep the denominator factorized for now. The probability of getting a green and then yellow is six 𝑥 minus 36 over 𝑥 multiplied by 𝑥 minus one.

To find the probability of getting a yellow counter and then a green, we multiply the probabilities for these branches of the tree diagram, giving 𝑥 minus six over 𝑥 multiplied by six over 𝑥 minus one. If we expand the bracket in the numerator, we have six 𝑥 minus 36 and we’ll keep the denominator factorized for now. In fact, we see that the probability of getting a green and then a yellow is the same as the probability of getting a yellow and then a green. We can have either a green and then a yellow or a yellow and then a green.

When we want to find the probability of one outcome or another, we add the individual probabilities together. So we’re adding together two lots of six 𝑥 minus 36 over 𝑥 multiplied by 𝑥 minus one. As these fractions have the same denominator, we add together the numerators. Six 𝑥 plus six 𝑥 is 12𝑥 and negative 36 plus negative 36 is negative 72, giving 12𝑥 minus 72 over 𝑥 multiplied by 𝑥 minus one.

Now, this would be an acceptable form in which to give our answer. But we could also spot that we can factorize the numerator by a factor of 12. In doing so, we, therefore, have a factorized form of an expression for the probability that Georgia takes one green and one yellow counter from the bag: 12 multiplied by 𝑥 minus six over 𝑥 multiplied by 𝑥 minus one.

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