What is the chane of this event happening?November 6, 2009 9:43 AM   Subscribe

I have an event that has a 75% chance of happening. If I run the trial seven times, what is the probability of the event happening at least once? And what's the math behind it?
posted by jackypaper to Science & Nature (7 answers total) 2 users marked this as a favorite

The probability of happening at least once is one minus the probability of not happening at all. The chance of not happening in one trial is 25%. The chance of not happening in two trials is 0.25 * 0.25 == 6.25%. And so on. The chance of not happening in 7 trials is 0.25 ^ 7, so very very low.

Google for the "birthday paradox".
posted by madmethods at 9:46 AM on November 6, 2009

For "at least once" flip the question around and ask yourself "What's the chance the event doesn't happen 7 times in a row?" If the event has a 25% chance of not happening, you can figure out what the odds are it doesn't happen 7 times in a row. Happening at least once would be anything but that, so 100% - the chance you just figured out.

Math: .25 ^ 7 = .000006, so = .999994, 99.994% or so.

I am currently sitting in statistics class discussing this exact same thing.
posted by true at 9:49 AM on November 6, 2009 [1 favorite]

It's 99.994%, or 1-(.25^7).
posted by deadmessenger at 9:50 AM on November 6, 2009

99.994% rather. Dropped a 9.
posted by true at 9:50 AM on November 6, 2009

Thanks! Perfect.
posted by jackypaper at 9:53 AM on November 6, 2009

You can think of this in terms of flipping a coin, except that the coin you are flipping is bad because it comes up heads three times out of four flips, and tails one time out of four.

The probability model that describes coin flipping is the Binomial distribution.

In terms of the Binomial as it is described on Wikipedia, you can think of getting a head on a coin toss as a "success", and getting a tail as a "failure".

Let's recast your question in these terms.

You have a coin that when flipped will give you a head 75% of the time and a tail 25% of the time. If you flip the coin seven times, what is the likelihood of getting at least one head ("success")?

If you flip the coin seven times, there are lots of possible events:

H T T T T T T
T H T T T T T
...
T T T T T T H
H H T T T T T
T H H T T T T
H T H T T T T
and so on...

That's a lot of events.

Let's look at the simpler case where you get all tails:

T T T T T T T

Whatever the probability is of getting all tails, if you subtract 1 from that probability, you get the probability of not getting all tails.

This is the same as the probability of getting one or more head on the coin toss.

We can use the Binomial to calculate the chance that you flip the coin and get seven tails in a row.

Here's the Binomial:

P(K = k successes) = (n choose k) p^k (1 - p)^(n-k)

The chance of getting a tails is 25%, so p = 0.25.

Let's write it out:

P(K = 7 tails) = (7 choose 7) (0.25)^7 (1 - 0.25)^(7-7) = (1) (0.25)^7 (1) = 0.000006

Therefore, 0.000006 is the probability of getting all tails on seven coin tosses.

Therefore, 1 - 0.000006 = 0.999994 is the probability of getting at least one or more heads from seven coin tosses.

You can use the Binomial to calculate this strictly in terms of getting heads, but the probability of getting one or more heads is the sum:

P(K = 1 head) + P(K = 2 heads) + ... + P(K = 7 heads)

which is more calculation. That's why it is easier to look at this in terms of getting all tails.
posted by Blazecock Pileon at 10:58 AM on November 6, 2009 [4 favorites]

Blazecock Pileon, in my probability and statistics class, we are allowed to bring in note cards to our exams and this next exam is covering, among other things, the binomial distribution. Somehow, I will fit your post onto a 3x5 card and bring it in.

Thanks for the A!
posted by fireoyster at 2:33 PM on November 6, 2009

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