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Hi sir pls help thanks in advance

My friend and I are in a debate based on a problem containing probability.

what is the probability of guessing the correct out come of 14 football matches (28 unique teams) with 3 possible outcomes in each match (Win draw and loss)

my logic says it is 1/3^(14),

but he debates it is [(1/3)*(2/3)*(2/3)]^14

can you please tell me the solution.

Assuming they all have equal chance (which they don't, but we'll ignore that for now),

each has a probability of 1/3. That is, 1/3 the time is a win, 1/3 the time is a loss,

and 1/3 the time is a draw.

Since there are 14 games, and each game has a 1/3 chance of being correct on the guess,

it is (1/3)^14.

The number 2/3 is the chance of guessing a game incorrectly.

That means the chance of guessing each game incorrectly is (2/3)^14.

If that's good, that's enough for you.

To give an even better understanding, though, read on...

The chance of guessing n games correctly (where n is an integer between 0 and 14) is given by

[(n choose 14)][(1/3)^n][(2/3)^(14-n)].

To say what "a choose b" means, it is a!/(b!(a-b)!) where '!' is used for factorial.

By this, 1! = 1;

2! = 2*1! = 2;

3! = 3*2*1 = 3*2! = 3*2 = 6;

4! = 4*3*2*1 = 4*3! = 4*6 = 24;

5! = 5*4*3*2*1 = 5*4! = 5*24 = 120;

6! = 6*5! = 6*120 = 720;

7! = 7*6! = 7*720 = 5,040;

8! = 8*7! = 40,320;

9! = 9*8! = 362,880;

10! = 10*9! = 3,628,800;

11! = 11*10! = 39,916,800;

12! = 12*11! = 479,001,600;

13! = 6,227,020,800; and

14! = 87,178,291,200.

Now if, for example, you only wanted 4 guesses to be correct, it would be the following:

Putting in 4 for x gives (14 choose 4)[(1/3)^4][(2/3)^10].

Now 14 choose 4 would be 87,178,291,200/(24*3,628,800).

Rather than doing it this way, a better way would be to note that 14!/10! = 14*13*12*11,

That makes the chance of guessing 4 correctly be (24,024/24)[(1/3)^4][(2/3)^10].

Note that 24,024/24 = 1,001, (1/3)^4 = 1/81 and (2/3)^10 = 1,024/59,049.

This means that the answer is given by 1,001(1/81)(1,024/59,049).

That comes down to you 1,025,024/4,782,969 = 0.214307055.

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Comment | Thanks soo much sir ..u de best!! |

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