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5 E) Predicting Outcomes – Part 1
5 E) Predicting Outcomes – Part 1
If we repeat an event a certain number of times, we are able to find the expected number of times that each of the different outcomes will occur. We do this by multiplying the probability of the event happening by the number of times that we repeat the event. This will make a lot more sense when we go through some examples.
Example 1
Suppose that we are flipping a coin 40 times. How many times would you expect the coin to land on heads?
There are two outcomes when we flip a coin, which are heads and tails. These outcomes are equally likely to occur, and this means that the probability of getting a head when flipping a coin is 0.5 or a ½. We are flipping the coin 40 times. We can work out the expected number of times that the coin will land on a head by multiplying the probability of the coin landing on a head by the number of times that we are flipping the coin.
Suppose that we are flipping a coin 40 times. How many times would you expect the coin to land on heads?
There are two outcomes when we flip a coin, which are heads and tails. These outcomes are equally likely to occur, and this means that the probability of getting a head when flipping a coin is 0.5 or a ½. We are flipping the coin 40 times. We can work out the expected number of times that the coin will land on a head by multiplying the probability of the coin landing on a head by the number of times that we are flipping the coin.
Therefore, we would expect the coin to land on heads 20 times out of the 40 times that the coin was flipped.
Example 2
We have a bag that contains 12 balls that are 3 different colours. The 3 different colours are red, green and yellow. The bag contains 6 green balls and 2 yellow balls. We are going to pick a ball out of the bag and replace the ball 60 times (we are going to note the colour of the ball and then place the ball back into the bag).
Part 1
The question gives the number of green and yellow balls, but not the number of red balls. We know that the total number of balls in the bag will equal the number of red balls, green balls and yellow balls added together. This gives us the equation below:
We have a bag that contains 12 balls that are 3 different colours. The 3 different colours are red, green and yellow. The bag contains 6 green balls and 2 yellow balls. We are going to pick a ball out of the bag and replace the ball 60 times (we are going to note the colour of the ball and then place the ball back into the bag).
- What is the probability of picking each of the different colours? Give your answers as fractions.
- What is the expected number of times that we will pick each of the three colours out of the bag?
Part 1
The question gives the number of green and yellow balls, but not the number of red balls. We know that the total number of balls in the bag will equal the number of red balls, green balls and yellow balls added together. This gives us the equation below:
The question tells us that there are 12 balls in total, 6 green balls and 2 yellow balls. We can sub these values into the above equation.
The first step in finding the number of red balls is to collect the two numbers on the right side of the equation.
The next step is to move the 8 from the right side of the equation to the left, which we are able to do by taking 8 from both sides of the equation.
We now know how many balls of each colour there are in the bag; there are 4 red balls, 6 green balls and 2 yellow balls. From this information, we can work out the probability of picking a particular coloured ball. We work out the probability of a particular coloured ball by using the following formula:
When we use this equation, we see that the probability of the colours are as follows (we need to remember to simplify the probabilities if we are giving them as fractions):
These probabilities are will be the same for every time that we pick a ball out of the bag. This is because we are picking a ball out, noting the colour and then replacing the ball in the bag. By replacing the ball that we have picked, the total number of balls and the number of each colour ball in the bag remains constant. This means that the probabilities of picking each of the coloured balls remains constant.
Part 2
We are now able to work out the expected number of times that a particular coloured ball will be picked by multiplying the probability of a particular colour by the number of times that we pick a ball out of the bag (which is 60).
Let’s first work out the expected number of times that we will pick a red ball.
Part 2
We are now able to work out the expected number of times that a particular coloured ball will be picked by multiplying the probability of a particular colour by the number of times that we pick a ball out of the bag (which is 60).
Let’s first work out the expected number of times that we will pick a red ball.
The expected number of times that a red ball will be picked is 20.
Let’s do the same process but for green balls.
Let’s do the same process but for green balls.
The expected number of times that a green ball is picked is 30.
And finally, let’s find the expected number of times that a yellow ball will be picked.
And finally, let’s find the expected number of times that a yellow ball will be picked.
Therefore, if we were to pick a ball out 60 times and replace the ball, we would expect to pick a red ball 20 times, a green ball 30 times and a yellow ball 10 times.