There are only red counters and blue counters in a bag. Joe takes at random a counter from the bag. The probability that the counter is red is 0.65 Joe puts the counter back into the bag. Mary takes at random a counter from the bag. She puts the counter back into the bag. (a) What is the probability that Joe and Mary take counters of different colours? There are 78 red counters in the bag. (b) How many blue counters are there in the bag?

GCSE Edexcel Maths Ratio Analysis and Problem Solving: There are only red counters and

There are only red counters and blue counters in a bag - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2

Question 17

There are only red counters and blue counters in a bag. Joe takes at random a counter from the bag. The probability that the counter is red is 0.65 Joe puts the cou... show full transcript

Worked Solution & Example Answer:There are only red counters and blue counters in a bag - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2

Step 1

What is the probability that Joe and Mary take counters of different colours?

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Answer

To find the probability that Joe and Mary take counters of different colors, we can consider the two possible scenarios:

Joe takes a red counter and Mary takes a blue counter.
Joe takes a blue counter and Mary takes a red counter.

Given the probability that Joe takes a red counter is 0.65, the probability that he takes a blue counter is:

$P(B) = 1 - P(R) = 1 - 0.65 = 0.35$

Since the selections are independent (Joe puts the counter back), the combined probability for both cases is:

For Joe red and Mary blue: $P(R ext{ and } B) = P(R) imes P(B) = 0.65 imes 0.35$
For Joe blue and Mary red: $P(B ext{ and } R) = P(B) imes P(R) = 0.35 imes 0.65$

Adding these two probabilities gives:

$P( ext{different colors}) = 0.65 imes 0.35 + 0.35 imes 0.65 = 2 imes (0.65 imes 0.35) = 0.455$

Thus, the probability that Joe and Mary take counters of different colors is 0.455.

Step 2

How many blue counters are there in the bag?

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Answer

We know there are 78 red counters, and the probability of selecting a red counter is 0.65. Let the number of blue counters be denoted as $B$ .

The total number of counters in the bag can be described as: $Total = 78 + B$

The probability that a counter picked is red is: $P(R) = \frac{78}{78 + B}$

Setting this equal to 0.65 gives: $\frac{78}{78 + B} = 0.65$

To solve for $B$ , multiply both sides by $(78 + B)$ : $78 = 0.65(78 + B)$

Expanding this equation: $78 = 0.65 imes 78 + 0.65B$

This simplifies to: $78 = 51.3 + 0.65B$

Rearranging gives: $78 - 51.3 = 0.65B$ $26.7 = 0.65B$

Dividing both sides by 0.65: $B = \frac{26.7}{0.65} \approx 41.0$

Since the number of counters must be whole, we can conclude that there are 42 blue counters in the bag.

There are only red counters and blue counters in a bag - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2

Worked Solution & Example Answer:There are only red counters and blue counters in a bag - Edexcel - GCSE Maths - Question 17 - 2018 - Paper 2

What is the probability that Joe and Mary take counters of different colours?

How many blue counters are there in the bag?

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