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(male narrator) So now we're gonna compute
some probabilities using counting.
So...a 4-digit PIN number is selected.
What is the probability that there are no repeated digits?
Uh...so that would mean that all the digits have to be different.
Uh...so if I was going to, uh...pick, uh...4 digits
where they're... all the digits are different,
I would have, uh...10 choices for the first digit--
assuming we're allowing values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
We have 10 choices for the first digit;
9 for the second digit,
because we're not gonna allow repeats;
uh...8 for the third digit; and 7 for the fourth digit--
or in other words, uh...that's 10P4.
Right?
Uh...is 5,040 different PINS... PINS...with...no...repeats.
Now in order to compute the probability,
we're gonna have to find the total number
of-of-of PIN numbers, including repeats.
So all PINS.
So if we consider all PINS that--
including allowing repeats--
how many choices are there for the first number--10;
how many for the second number--
also 10, also 10, also 10.
Because we allow repeats, so there's no exclusions.
So that's 10 to the 4th,
and 10 to the 4th is...uh...10,000.
So there's 10,000 total PINS.
Out of those, 5,040 of them have no repeats.
So we got 5,040 out of 10,000-- or about 50.4% of PINS--
uh...have no repeated digits.
So now let's look at a-a lottery problem.
Uh...so in a certain state's lottery,
we got 48 different numbers,
uh...and 6 of them are drawn at random.
Uh...and if you have... and you...
the player gets to choose 6 numbers,
and if their 6 numbers match, they win a $1 million.
And the order doesn't matter in this particular lotto,
so we're not playing Powerball.
[laughs] Uh...and let's see if we...
let's see if we can find the probability that...
we win that, um...we win that million dollar prize
if we purchase a single ticket.
So the first thing we didn't know
is how many possible outcomes are there?
So from 48 different numbers,
uh...they're...we're going to choose 6 of them.
And order here does not matter,
uh...which is why we're using combinations.
And so we end up
with, uh...12,271,512 different 6-number outcomes.
Now if you have one ticket, then you have one outcome,
and so your probability of winning
is 1 out of 12,271,512,
which is a really, really small number.
Uh...and so that's the probability
of winning the main prize.
Now oftentimes, there's a second prize,
and in this case, uh...it's a $1,000
if you can match 5 of the numbers.
Now there are still 48 choose 6,
uh...12,271,512 different outcomes,
but now we need to figure out how many of those involve
us winning, uh...matching 5 of the numbers.
Now in order for that to happen, from our 6 numbers on the card,
we're gonna have to match 5 of them.
So how many different ways
can we pick 5 matching numbers out of our 6?
That would be 6 choose 5, which is 6.
Now we also have to consider that, you know,
how many different ways could we not pick the-the last number?
Uh...so from the 42 non-winning numbers,
uh...we need to pick 1 of them to be on our card,
and so that is-is 42.
So our final probability of winning is gonna be...
there are...uh... that many--
42 times 6 is 252 different ways
that we can match 5 numbers and not match the 6,
out of our, uh...48 choose 6,
out of our $12,271,512 total outcomes.
Uh...and that gives us
a, you know, marginally higher probability of winning--
though it's still pretty small.