Video Transcript
Suppose 𝐴 and 𝐵 are two events in
a random experiment. Given that the probability of 𝐴
union 𝐵 equals 0.68 and that the probability of 𝐴 intersect 𝐵 is equal to 0.12
and the probability of 𝐴 intersect 𝐵 is equal to the probability of 𝐴 times the
probability of 𝐵, what are the possible values for the probability of 𝐴 and the
probability of 𝐵?
Because we’re given that the
probability of 𝐴 intersect 𝐵 is 0.12, we know that these events are not mutually
exclusive. They can happen at the same
time. And we know for non-mutually
exclusive events, the probability of 𝐴 union 𝐵, that’s the probability that 𝐴 or
𝐵 occurs, is equal to the probability of 𝐴 plus the probability of 𝐵 minus the
probability of the intersection of 𝐴 and 𝐵.
In our question, the probability of
the union of 𝐴 and 𝐵 is 0.68. Probability of 𝐴 and 𝐵 are
unknown. But we do know the probability of
the intersection of 𝐴 and 𝐵 is 0.12. We can simplify this expression a
little bit by adding 0.12 to both sides of the equation. This shows us that the probability
of 𝐴 plus the probability of 𝐵 is equal to 0.80. This gives us one equation to work
with. We’ve also been told that the
probability of the intersection of 𝐴 and 𝐵 is equal to the probability of 𝐴 times
the probability of 𝐵. Since we know that this probability
of the intersection of 𝐴 and 𝐵 is 0.12, we now have a second equation we can use
to solve for the probability of 𝐴 and 𝐵. And that is that the probability of
𝐴 times the probability of 𝐵 must be equal to 0.12.
Since we’re going to be solving
some simultaneous equations, it’s probably easier if we make some substitutions for
our variables. Instead of having to write the
probability of 𝐴 over and over again, we’ll just use the variable 𝑎 to represent
this and lowercase 𝑏 to represent the probability of 𝐵, which means equation one
now says 𝑎 plus 𝑏 is equal to 0.8 and equation two says 𝑎 times 𝑏 is equal to
0.12. We can solve for the 𝑎-variable in
equation one by subtracting 𝑏 from both sides of the equation so that 𝑎 is equal
to 0.8 minus 𝑏. And then we’ll substitute the value
0.8 minus 𝑏 in for 𝑎 in our second equation.
To solve, we’ll have to distribute
this 𝑏 over the 0.8 and the negative 𝑏, which gives us 0.8𝑏 minus 𝑏 squared is
equal to 0.12. And now, all of a sudden, we have a
square term in our equation, which makes this a quadratic equation and tells us
we’ll be finding two different values for 𝑏. To solve a quadratic equation, we
wanna set this equal to zero. We do this by subtracting 0.8𝑏
from both sides of the equation and adding 𝑏 squared to both sides of the
equation. We then have zero equals 𝑏 squared
minus 0.8𝑏 plus 0.12. At this point, it’s completely fine
to use the quadratic formula.
However, there is one strategy we
can try for factoring. If we rewrite negative 0.8 as
negative four-fifths, its fraction equivalent, and we rewrite 0.12 as its fractional
equivalent, three twenty-fifths, we can break our 𝑏 squared up into 𝑏 times 𝑏 and
then consider what two fractional values multiply together to equal three
twenty-fifths and add together to equal negative four-fifths. Since three is a prime number,
we’re looking for fractions that have a one and a three in their numerator. And we know that five times five
equals 25, one-fifth times three-fifths does equal three twenty-fifths, and
one-fifth plus three-fifths equals four-fifths. We need negative four-fifths, which
means we can make both of these values negative.
Our 𝑏-values will be the values
that make each of these terms equal to zero. 𝑏 would equal one-fifth or 𝑏
would equal three-fifths. Since the probabilities we began
with were given in decimal form, we can rewrite this to say 𝑏 equals 0.2 or 𝑏
equals 0.6. We then go back to our first
equation. If we plug in 0.2 for 𝑏, we know
that 𝑎 plus 𝑏 combined is 0.8. So we can subtract 0.2 from both
sides of this equation, and we get that 𝑎 equals 0.6 when 𝑏 equals 0.2. Because we haven’t been given any
other information about 𝑎 and 𝑏, we have to say that the other option is when 𝑎
is 0.2 and 𝑏 is 0.6.
At this point, we need to go back
and use the notation we started with, the probability of 𝐴 and the probability of
𝐵. And we’ll say the probability of 𝐴
is equal to 0.2 and the probability of 𝐵 is equal to 0.6 or the probability of 𝐴
equals 0.6 and the probability of 𝐵 is equal to 0.2. Both of these combinations fit that
the probability of 𝐴 and 𝐵 must combine together to equal 0.8 and must multiply
together to equal 0.12.
