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## Precalculus

### Course: Precalculus > Unit 8

Lesson 5: Probability using combinatorics- Probability using combinations
- Example: Lottery probability
- Example: Different ways to pick officers
- Probability with permutations & combinations example: taste testing
- Probability with combinations example: choosing groups
- Probability with combinations example: choosing cards
- Probability with permutations and combinations
- Mega millions jackpot probability

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# Mega millions jackpot probability

Probability of winning the Mega Millions jackpot. Created by Sal Khan.

## Want to join the conversation?

- Since the winnings are much greater than $176 million, doesn't this mean that it would be reasonable to buy 176 million tickets, guaranteeing that you make a profit of ~$300 million?(94 votes)
- An Australian investment group tried this a few years ago. They paid a lot of people to buy up a lot of tickets for them. They didn't buy enough to guarantee the jackpot win, but still made millions in profit on lesser prizes.(76 votes)

- How was the probability of being struck by a lightning calculated? Because 1/10,000 doesn't sound like a lot.(18 votes)
- National Weather Service did the calculation based on reports provided to them: http://www.lightningsafety.noaa.gov/medical.htm

ODDS OF BECOMING A LIGHTNING VICTIM

(based on averages for 2001-2010)

Estimated U.S. population as of 2011

310,000,000

Annual Number of Deaths Reported

39

Number of Injuries Reported

241

Estimated number of U.S. Deaths

40

Estimated number of actual Injuries

360

Odds of being struck by lightning in a given year (reported deaths + injuries)

1/1,000,000

Odds of being struck by lightning in a given year (estimated total deaths + injuries)

1/775,000

Odds of being struck in your lifetime (Est. 80 years)

1/10,000

Odds you will be affected by someone being struck (Ten people affected for every one struck)

1/1000(67 votes)

- Isn't odds the same as probability?(17 votes)
- While Theresa's answer is correct in an academic or technical context, the use of "odds" colloquially to mean the same as "probability" is
**very**common (as Sal did, and also http://www.sheknows.com/living/articles/1023453/what-are-the-odds-21-statistics-that-will-surprise-you, https://www.cbsnews.com/news/powerball-mega-millions-lottery-odds-us/, http://www.nsc.org/learn/safety-knowledge/Pages/injury-facts-chart.aspx ).

It's important when studying probability to be unambiguous, so it's good that Sal corrected that usage in the video. But in a non-technical context, you have to be prepared for the possibility that someone who says "odds" really**does**mean "probability". In fact that usage is so common that you would have an uphill battle making the case that the technically correct usage is the**only**correct usage.(5 votes)

- does buying more than one tickets increase (mathematically of course) a person's odds of winning. 1 vs 100 tickets. thanks(9 votes)
- Yes. Assuming all the tickets have different numbers, if you have 100 tickets, you have 100 times the chance of winning.(19 votes)

- Where can I find a video about which to choose: a lump sum or getting money over time?(6 votes)
- I'm not sure if there's one here, but it's a net present value calculation.(2 votes)

- If the odds of selecting a winning lotto number is 1 : 177M and a ticket is $1. Once the jackpot is over 177M + tax + interest on loan, why not take out a loan and buy every ticket?! Betting on the chance that you don't split the winnings with anyone this is a minimal risk "investment."(2 votes)
- Hmm. There is an old saying, the lottery is a tax on people who didn't think they would need algebra in real life.

Given how many tickets are sold when the lottery gets that large, there is an excellent chance of more than one winner.

BTW, you'll notice that they make it physically impossible to play all possible numbers. Even if you somehow managed to play 10 sets of numbers a second, round the clock, non-stop, it would take you over 200 days to play all of possible combinations, though most lotteries only give you a few days to buy tickets.

Thus, it absolutely impossible to buy every combination for the large pay-off lotteries. That is not by accident.(9 votes)

- So let's say I own a surfboard-making shop, and it's a very good shop. Only one in 1000 boards is messed up. I also have a test which is 99% accurate. If my surfboard tests bad, then what is the probability that it is bad? How would I go about doing something like this?(2 votes)
- Assuming that the boards are all tested independently and that problems making boards occur independently, a board that tests as being bad is 99% likely to actually be bad - the probability should be the same as the accuracy of the test.

It's slightly more complicated if you consider something like the probability of selling a bad board, which should be about 1 per 100,000 -- which is 1/1000 for the chance of it actually being bad, multiplied by the 1/100 chance of it wrongly passing the test. 99 bad boards from that 100,000 would be found and not sold (99/100 detected x 1/1000 bad), while around 100 good boards would test bad (1/100 wrong tests x 999/1000 good boards).(5 votes)

- Hi,

I'm just curious why you multiply by 46 and not (1/46)?

To my understanding, choosing the correct super ball from the bin of 46 would only be one possible outcome of a total possible 46 balls in the bin...(2 votes)- Hello, sorry your question has gone unanswered for so long. This is my understanding of the situation. You may not have realized that when Sal Khan multiplied by 46 he was working in the denominator. Sal multiplied the denominator by 46 which is essentially multiplying by 1/46. This is what Sal did:

This is the same thing as what Sal did, just combining them in a different order:`1 1`

—————————————— = ———————————

3,819,816 x 46 175,711,536

I hope this helps!`1 1 1`

————————— x —— = ———————————

3,819,816 46 175,711,536(6 votes)

- then how come a person has been struck by lightning 7 times(2 votes)
- Great Question.

Just because the probability of something is very very remote does not preclude it from ever happening. The guy that was hit 7 times, do you know what he did for a living? Perhaps it was something that put him at higher risk, like needing to be out doors and exposed even if a storm happens, like a park ranger or something, and possibly he works in an area that has a higher incidence of lightning storms due to its geographic location. All these are factors to consider.(5 votes)

- A related question: suppose I wanted to try to win the lottery by buying all the tickets (this isn't hypothetical, there are investment groups that actually do this!). Since there's no way to sequentially buy each number (manually filling out cards is too impractical), I resort to quickpicks, which means there is no guarantee that I'm not buying the same number multiple times.

So the question: if I set a target of obtaining 1/2 the available numbers, how much on average do I need to spend?(3 votes)

## Video transcript

I've been asked to calculate
the probability of winning the Mega Millions jackpot. So I thought that's what
I would do this video on. So the first thing is to
make sure we understand what does winning the
jackpot actually mean. So there's going to
be two bins of balls. One of them is going to have 56
balls in it, so 56 in one bin. And then another bin is
going to have 46 balls in it. So there are 46 balls in
this bin right over here. And so what they're
going to do is they're going to pick 5 balls
from this bin right over here. And you have to get the exact
numbers of those 5 balls. It can be in any order. So let me just draw them. So it's 1 ball-- I'll shade
it so it looks like a ball-- 2 balls, 3 balls, 4
balls, and 5 balls that they're going to pick. And you just have to get
the numbers in any order. So this is from a bin of 56. And then you have to
get the mega ball right. And then they're
going to just pick one ball from there, which
they call the mega ball. And obviously,
this is just going to be picked-- this is
going to be one of 46, so from a bin of 46. And so to figure out the
probability of winning, it's essentially
going to be one of all of the possibilities of numbers
that you might be able to pick. So essentially, all
of the combinations of the white balls times
the 46 possibilities that you might get
for the mega ball. So to think about
the combinations for the white balls, there's a
couple of ways you could do it. If you are used to thinking
in combinatorics terms, it would essentially say, well,
out of a set of 56 things, I am going to choose 5 of them. So this is literally, you could
view this as 56, choose 5. Or if you want to think of
it in more conceptual terms, the first ball I pick,
there's 56 possibilities. Since we're not replacing the
ball, the next ball I pick, there's going to be
55 possibilities. The ball after that, there's
going to be 54 possibilities. Ball after that, there's
going to be 53 possibilities. And then the ball
after that, there's going to be 52 possibilities,
because I've already picked 4 balls out of that. Now, this number right over
here, when you multiply it out, this is a number
of permutations, if I cared about order. So if I got that
exact combination. But to win this, you don't
have to write them down in the same order. You just have to get those
numbers in any order. And so what you
want to do is you want to divide this
by the number of ways that five things can
actually be ordered. So what you want to do
is divide this by the way that five things can be ordered. And if you're
ordering five things, the first of the five things can
take five different positions. Then the next one will have four
positions left, and then the one after that will have
three positions left. The one after that will
have two positions. And then the fifth one will
be completely determined because you've already
placed the other four, so it's going to have
only one position. So when we calculate this
part right over here, this will tell us all
of the combinations of just the white balls. And so let's calculate that. So just the white balls, we
have 56 times 55 times 54 times 53 times 52. And we're going to divide that
by 5 times 4 times 3 times 2. We don't have to multiply
by 1, but I'll just do that, just to show
what we're doing. And then that gives
us about 3.8 million. So let me actually let
me put that off screen. So let me write
that number down. So this comes out to 3,819,816. So that's the number
of possibilities here. So just your odds of picking
just the white balls right are going to be one
out of this, assuming you only have one entry. And then there's
46 possibilities for the orange balls,
so you're going to multiply that times 46. And so that's going
to get you-- so when you multiply it times
46-- bring the calculator back. So we're going to multiply
our previous answer times 46. "Ans" just means
my previous answer. I get a little
under 176 million. Let me write that number down. So that gives us 175,711,536. So your odds of winning it,
with one entry-- because this is the number of possibilities,
and you are essentially, for $1, getting one of
those possibilities. Your odds of winning is
going to be 1 over this. And to put this in a
little bit of context, I looked it up on
the internet what your odds are of
actually getting struck by lightning
in your lifetime. And so your odds
of getting struck by lightning in your
lifetime are roughly 1 in 10,000-- chance
of getting struck by lightning in your lifetime. And we can roughly say your odds
of getting struck by lightning twice in your lifetime,
or another way of saying it is the odds of
you and your best friend both independently being struck
by lightning when you're not around each other, is going
to be 1 in 10,000 times 1 in 10,000. And so that will get you 1 in--
and we're going to have now eight 0's-- 1, 2,
3, 4, 5, 6, 7, 8. So that gives you
1 in 100 million. So you're actually
twice-- almost, this is very rough--
you're roughly twice as likely to get struck by
lightning twice in your life than to win the Mega jackpot.