# Video: Pack 4 • Paper 3 • Question 12

Pack 4 • Paper 3 • Question 12

04:16

### Video Transcript

Anne, Jamie, and Miguel investigated a biased coin. They each flipped the coin multiple times and recorded the number of times the coin landed on heads and the number of times it landed on tails. They each used their own results to calculate a probability that the coin will land on tails when it is flipped once. Who will have calculated the best estimate for this probability? Give a reason to support your answer.

A coin is fair if both sides have the same probability of a half of showing during a flip. It is deemed to be biased when these probabilities are not equal. For our biased coin, each person has recorded their results in a table. We can use these tables to estimate probabilities. This is called experimental probability or relative frequency. The more results we have, the more accurate the relative frequency will be. So we’ll start by finding the total number of flips performed by each person.

Anne flipped the coin 15 plus 40, which is 55 times. Jamie flipped it 28 plus 82, which is 110 times. And Miguel flipped it seven plus 28, which is 35 times. We can see that Jamie flipped the coins the most times. Therefore, his estimate for the probability that the coin will land on tails will be the most accurate.

They combined all of their results. Use the combined results to calculate the probability that when the coin is flipped three times, it will land on heads the first two times and tails the third time.

Let’s begin by combining their results. 15 plus 28 plus seven is 50. So the coin landed on heads 50 times. 40 plus 82 plus 28 is 150. So the coin landed on tails 150 times. 50 plus 150 is a total of 200 flips.

The probability of an event occurring is given by the number of ways this event can occur divided by the total number of outcomes. In this case then, the probability that the coin lands on heads is 50 out of 200. We can simplify this by looking for the highest common factor of 50 and 200. This is 50. 50 divided by 50 is one and 200 divided by 50 is four. So the probability that the coin lands on heads is a quarter.

We can work out the probability that it lands on tails one of two ways. Using the formula, we get 150 over 200. Dividing both the numerator and the denominator by 50 gives us three-quarters. Remember though the probability of all possible outcomes of an experiment, which is here flipping the coin, adds to one. We could instead then have subtracted the probability of flipping a heads from one: one minus a quarter is three-quarters.

Either method is valid and will get you the marks in an exam. We’re being asked to find the probability that when it’s flipped three times, it will land on heads the first two times and tails the third. That’s the probability that it lands on heads and then on heads and then on tails.

When two events are independent, the probability that one and the other happens is calculated by multiplying their individual probabilities together. In this case, there are three events. We can multiply the three probabilities together. That’s a quarter multiplied by a quarter for the two heads multiplied by three-quarters for the final tails. One times one times three is three and four multiplied by four multiplied by four is 64.

The probability when the coin is flipped three times that it will land on heads the first two times and tails the third is three out of 64.