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
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
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
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.