## Friday, October 13, 2006

### Probability For Dummies

Every so often someone asks a question like, "What's the probability of getting at least one 6 rolling two dice?" A common answer is 1/3, but this is not correct. The thinking here is that there's a 1/6 probability of rolling a 6 on each die, and 1/6 + 1/6 = 1/3. So I thought I'd talk about it.

The probability of any event occurring ranges from 0 (will not occur) to 1 (will occur). Probabilities can be expressed as a fraction, a decimal value, or a percent (which really just a decimal times 100).

The most common kinds of randomness in games are rolling dice and drawing cards from a shuffled deck. In both of these cases, the results are independent. This is a very important concept.

An independent event occurs without relation to any other event. For example, the results of any die roll are not affected by any past die rolls. This means that if you roll ten 6's in a row, the probability of rolling another 6 is still 1/6.

The other basic assumption is that the randomizing mechanism is fair. In the case of a 6-sided die, it is assumed to be balanced in such a way that the probability of each result is equal. This does not mean that if you roll 1,2,3,4,5 that the next roll will be a 6. It means that over a very large number of rolls, the number of each will be very close. The assumption also holds for cards. A well-shuffled deck implies that each card has an equal chance to be on the top.

To calculate the probability of a single fair event, divide the number of desired outcomes by the number of possible outcomes.

For example, the probability of rolling exactly a 3 on a d6 is 1 (one outcome in question) divided by 6 (six possible outcomes) making 1/6. This is probably intuitive for most people.

The probability of rolling an odd number is 3 (1, 3, 5) divided by 6, which is 3/6 or 1/2. This true of any 3 possible outcomes (rolling an even number, rolling 3 or less, rolling 2 3 or 5, etc.)

Now for more complex stuff.

The probability of multiple independent events ALL occurring is the product of their individual probabilities. However, you must be careful with the exact meaning of what you are looking for.

What is the probability of rolling two 6's on 2d6? The probabilities for each are 1/6. 1/6 * 1/6 = 1/36 (or 2.77%). Easy!

What is the probability of rolling 3 or more (both times) on 2d6? The probabilities for each are 4/6 (or 2/3). 2/3 * 2/3 = 4/9 (or 44.44%).

What is the probability of rolling a 6, then rolling an even number? 1/6 * 1/2 = 1/12 (or 8.33%).

What is the probability of rolling a 1 and a 5? This is the kind of question where you can easily go wrong. Do you mean rolling a 1 first, and rolling a 5 second? If so, it's 1/6 * 1/6 = 1/36 (or 2.77%). If you mean rolling two dice and having a 1-5 appear then what you really mean is rolling a 1 then a 5, or rolling a 5 then a 1. This leads us to:

The probability of one of several independent events occurring is the sum of their individual probabilities.

In the previous example, the probabilities of both 1-5 and 5-1 rolls are 1/36, so the probability of either of them occurring is 1/36 + 1/36 = 2/36 or 1/18 (or 5.55%).

What is the probability of rolling an odd/even pair on 2d6? This is a similar question. The probability of rolling odd then even is 1/2 * 1/2 = 1/4. The same for even-odd. So the total is 1/4 + 1/4 = 1/2 (or 50%).

Now let's kick it up a notch.

What is the probability of rolling 9 or more on 2d6? Rather than iterating all the probabilities for 9's, 10's 11's and 12's, you can do this by making a table:

___1__2__3__4__5__6
1__2__3__4__5__6__7
2__3__4__5__6__7__8
3__4__5__6__7__8__9
4__5__6__7__8__9_10
5__6__7__8__9_10_11
6__7__8__9_10_11_12

You can see that 10 of the 36 results are 9 or more. And, as you guessed, the probability is 10/36 or 5/18 (or 27.77%).

What is the probability of rolling doubles (not triples) on 3d6? This one is tricky. Let's break it down by first examining case one: the first two rolls match and third does not. The first roll can be anything (probability 1). The second roll must match the first (probability 1/6). The third roll must not match (probability 5/6, do you see this?). So to total probability for case one is 1 * 1/6 * 5/6 = 5/36.

The second case is the same as the first one, but rearranged: any roll, no match, match first roll. Again we have 1 * 5/6 * 1/6 = 5/36. Likewise with the third case: anything, no match, match second roll = 1 * 5/6 * 1/6 = 5/36.

The final answer is the sum of the three separate cases: 5/36 + 5/36 + 5/36 = 15/36 (41.66%).

What is the probability of rolling three completely different numbers on 3d6? This one should be easy by now. The first number can be anything (1), the second must be anything but the first (5/6), and the third must be anything but the first two (4/6 or 2/3), resulting in 1 * 5/6 * 2/3 = 10/18 or 5/9 (or 55.55%).

Interesting note. Look at the previous two examples. One is for none of the three rolls to match, and other is for two of the rolls to match. The only other possibility is 3-of-a-kind. 1 (36/36) - 5/9 (20/36) - 15/36 = 1/36, which is the answer! The sum of the probabilities of all possible outcomes is 1.

What is the probability of drawing 2 aces from the top of a shuffled standard deck of cards? In this case, the second drawn card is going to be affected by the first drawn card, because the number of cards remaining in the draw pile has been changed. There are 4 aces in the deck and 52 cards. The probability of the first card drawn being an ace is therefore 4/52 or 1/13. Now what's left? There are 3 possible aces out of 51 cards: 3/51 or 1/17. So the answer is 1/13 * 1/17 = 1/221 (or 0.45%).

Now back to the original question. What's the probability of getting at least one 6 rolling two dice? Can you do it? There are two cases: 6-anything, and anything-6. However, notice that these two cases overlap. To make them distinct, we need to change the wording a bit: 6-anything, and not6-6. Case one is 1/6 * 1 = 1/6. Case two is 5/6 * 1/6 = 5/36. The final answer is 1/6 (6/36) + 5/36 = 11/36 (or 30.55%)!

At 2:25 PM,  Mark Haberman said...

You could also say that the probability of rolling at least one six is 1 - the probabilty of rolling no 6's

or
1 - (5/6 * 5/6) = 30.5%

At 2:27 PM,  ekted said...

Yes. Amazing how it works out to the same answer from both directions. :)

Since when did the probability of drawing cards from a shuffled deck become classified as an independent event?

Surely you meant the first card only?

At 6:09 AM,  ekted said...

Yes, drawing *a* card is independent. Once you remove the top card, drawing the next card is independent with respect to the cards left (although the composition of the deck is dependent upon the first card drawn). Perhaps I tread on semantics. Probability is not my forte.

At 7:40 PM,  MrNw said...

I failed statistics at University, but surely the odds of getting at least one 6 on 2 dice = 1/6 +1/6 = 2/6 = 33.33%.

I think the question you are answering is:
What's the probability of getting only one 6 rolling 2 dice?

At 7:59 PM,  ekted said...

Make a 6x6 table of all possible combinations. You will see that 11 of the 36 cells have one or more 6's. That's 11/36 (or 30.55%).

The PROBABILITY of rolling exactly one 6 on two dice is 10/36 (or 27.77%).

At 8:52 PM,  MrNw said...

You are correct.

The double 6 result only counts once, not once for each set of possibilities. No wonder I failed, the correct answer just did not feel right.

At 7:47 PM,  Anonymous said...

Hey if you are interested, there's a real Probability for Dummies book if you want to check it out.

At 5:35 PM,  Anonymous said...

What's the probability of rolling a triple with 3 dice? 1/36?

At 5:43 PM,  ekted said...

Yes, any triple is 1/36.

At 2:06 PM,  Anonymous said...

What is the probabiltity of rolling double three times?

At 4:54 PM,  ekted said...

1/6 * 1/6 * 1/6 = 1/216 or 0.46%

At 8:18 PM,  Anonymous said...

What is the theoretical probability of rolling three doubles in a roll if you roll two six sided die

At 12:50 AM,  Anonymous said...

1/6x1/6x1/6=1/216

At 12:11 AM,  hswiseman said...

This comment has been removed by the author.

At 12:15 AM,  hswiseman said...

My Mother has two daughters and six natural granddaughters. Each of these female offspring/grand-offspring have birthdays that are the either the 5th of the month or multiples of 5.

DAUGHTERS
A 9/05
T 3/30
GRANDDAUGHTERS
S 9/15
N 8/15
C 9/10
B 1/15
J 2/15
S 12/25

Could you tell me the odds of this occurring?

71/365 days of the year are 5's or multiples thereof. Let's assume that with each birth, there was a 50/50 shot of being born a girl. 8/8 ocurrences happened on the 5's.

At 5:11 PM,  Anonymous said...

I'm pretty sure the chance to roll a straight (1,2,3 or 3,4,5, etc.) on 3d6 is 1/18. But if I'm trying to get a 3-number straight on 4d6 what is the probability? I keep getting lost trying to figure it out.

At 7:35 AM,  Anonymous said...

What are the chances of a single match on 4d6. And on 5d6?

For example, rolling two 3's on 5d6?

At 10:52 PM,  Anonymous said...

What is chance of rolling 3 fours on one roll if using three four sided die.

At 1:11 PM,  Anonymous said...

What is the probability of rolling a single "1" (no snake eyes)on a pair of dice? Would it be 5/36 or 10/36 (for reversing the combo's)?
Ex: a 1-2 is different from a 2-1?

Also, is it possible to reverse the combinations when considering doubles? (Example: there is technically 6/36 doubles OR could it be considered 12/36 if you reverse the #'s order)

At 1:31 PM,  ekted said...

Either you roll a 1 followed by not-a-1, or not-a-1 followed by a 1. So 1/6 * 5/6 + 5/6 * 1/6 = 10/36.

At 3:47 PM,  Anonymous said...

When rolling a d6 ten times what is the probability of rolling the same amount of even numbers as odd numbers?

At 2:17 PM,  Anonymous said...

What's the probability of rolling six 4's on one dice of six sides???

At 5:09 AM,  Anonymous said...

I love Hattie Jana Ord

At 5:15 AM,  Anonymous said...

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At 5:17 AM,  Anonymous said...

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At 5:27 AM,  Alishba khatri said...

Thanks Very Nice Post Regards: Techrainy

At 12:29 PM,  Anonymous said...

What is the probability of rolling 5 4's with 5 6-sided dice?