Video Transcript
Mason and Ethan are using their
computers to take part in an online experiment on a website. When Mason presses the space bar on
his keyboard, there’s a 50-percent probability that his screen turned blue. When Ethan presses the space bar on
his keyboard, there’s a 45-percent probability that his screen turns blue. If they both press their space
bars, there’s a 15-percent chance that both of their screens turn blue. Are “Mason’s screen turning blue”
and “Ethan’s screen turning blue” independent events?
We’re asked in this question to
work out whether two events are independent or not. That is, are the events “Mason’s
screen turning blue” and “Ethan’s screen turning blue” independent or not? So let’s first remind ourselves
what we mean when we say two events are independent. We say that two events 𝐴 and 𝐵
are independent if the fact that 𝐴 occurs does not affect the probability of 𝐵
occurring, and vice versa. And for independent events 𝐴 and
𝐵, the probability of 𝐴 and 𝐵 is equal to the probability of 𝐴 times the
probability of 𝐵.
But let’s look at the formula now
for conditional probability. This says that the probability of
𝐴 given that 𝐵 has occurred is the probability of 𝐴 and 𝐵 over the probability
of 𝐵. We can rearrange this, multiplying
both sides by 𝐵 so that we have the probability of 𝐵 times the probability of 𝐴
given 𝐵 is equal to the probability of 𝐴 and 𝐵. Swapping sides, we can compare this
to our first equation for independent events. And we can see that if 𝐴 and 𝐵
are independent events, then the probability of 𝐴 given 𝐵 must equal the
probability of 𝐴. And the same idea applies for the
probability of 𝐵. That is, the probability of 𝐵
given that 𝐴 has occurred is equal to the probability of 𝐵.
And this makes perfect sense
because if 𝐴 and 𝐵 are independent, the fact that 𝐴 has occurred does not affect
the probability of 𝐵 occurring. And this gives us a really useful
way of testing whether two events are independent or not. So let’s apply this now to our
question with Mason and Ethan.
Let’s let 𝑃 𝑀 be the probability
that Mason’s screen turns blue and 𝑃 𝐸 be the probability that Ethan’s screen
turns blue. We’re told that when Mason presses
the space bar on his keyboard, there’s a 50-percent probability his screen turns
blue. That is, 𝑃 𝑀 is 50 percent. And when Ethan presses the space
bar on his keyboard, there’s a 45-percent probability that his screen turns blue so
that 𝑃 𝐸 is 45 percent. We can write these as fractions, 50
out of 100 and 45 out of 100, or as decimals, that’s 0.5 and 0.45. We’re also told that if they both
pressed their space bars, there’s a 15-percent chance that both of their screens
turned blue. And that’s 0.15 as a decimal so
that the probability of 𝑀 intersection 𝐸, that’s both 𝑀 and 𝐸 occurring, is
0.15.
Now we want to know whether
“Mason’s screen turning blue” and “Ethan’s screen turning blue” are independent
events. And we know that for two
independent events, our two equations must both be true for probability of 𝐴 given
𝐵 is equal probability of 𝐴 and probability of 𝐵 given 𝐴 is the probability of
𝐵. And if our two events are
independent, then the probability of 𝑀 given 𝐸 should equal the probability of 𝑀
and the probability of 𝐸 given 𝑀 should be equal the probability of 𝐸. So let’s call these equations one
and two and see if these equations are true using the conditional probability
formula with our values.
By the conditional probability
formula, the probability of 𝑀 given 𝐸, that’s the probability of Mason’s screen
turning blue given that Ethan’s screen turns blue is the probability of 𝑀 and 𝐸
over the probability of 𝐸. That is the probability of both
screens turning blue over the probability of Ethan’s screen turning blue. We know that our numerator is 0.15
and that our denominator is 0.45 so that the probability of 𝑀 given 𝐸 is 0.15 over
0.45. And this evaluates to 0.3
recurring.
If you haven’t seen this notation
before, the dot above the three means that the three is repeated indefinitely. So we have that the probability of
𝑀 given 𝐸 is 0.33 recurring. And if 𝑀 and 𝐸 are independent
events, this should equal the probability of 𝑀. But the probability of 𝑀 is 0.5,
which is not equal to 0.3 recurring. So, in fact, our equation one is
not true. The probability of 𝑀 given 𝐸 is
not equal to the probability of 𝑀.
So now let’s look at our second
equation, the probability of 𝐸 given 𝑀, does this equal the probability of 𝐸? The probability of 𝐸 given 𝑀, by
our conditional probability formula, is the probability of 𝐸 and 𝑀 over the
probability of 𝑀. That is the probability of Ethan’s
screen turning blue given that Mason’s screen turned blue is the probability of both
turning blue over the probability of Mason’s screen turning blue. Using our values, we have 0.15 over
0.5, since the probability of 𝑀 and 𝐸 is the same as the probability of 𝐸 and 𝑀,
since multiplication is commutative. And 0.15 over 0.5 evaluates to
0.3. And this does not equal the
probability of Ethan’s screen turning blue, which is 0.45. So in fact, our second equation is
not true either. The probability of 𝐸 given 𝑀 does
not equal the probability of 𝐸.
Since neither of the required
equations are true, the two events, Mason’s screen turning blue and Ethan’s screen
turning blue, are not independent. This means that the probability of
Ethan’s screen turning blue is affected by what happens with Mason’s screen, and
vice versa. What occurs on one of the screens
affects the probability of the other screen turning blue.
