Coin Tossing Probability?

Question

If 10 Fair coins are tossed all at once, whats the probability that 5 will be heads and 5 will be tails?

Answer 1

Pascal's Triangle (named after Blaise Pascal, (June 19, 1623–August 19, 1662, a French mathematician, physicist, and religious philosopher) tells you.

...................1 1
.................1 2 1
...............1 3 3 1
.............1 4 6 4 1
...........1 5 10 10 5 1
,........1 6 15 20 15 6 1
.......1 7 21 35 35 21 7 1
.....1 8 28 56 70 56 28 8 1
...1 9 36 84 126 126 84 36 9 1
.1 10 45 120 210 252 210 120 45 10 1

The nth row is the number of ways that you can expect n fair coins will be tossed in the long term.

Thus in the tenth row there are 2^10 = 1024 outcomes for ten tosses:

1 way of getting 10 heads and 0 tails
10 ways of getting 9 heads and 1 tail
45 ways of getting 8 heads and 2 tails
120 ways of getting 7 heads and 3 tails
210 ways of getting 6 heads and 4 tails
252 ways of getting 5 heads and 5 tails
210 ways of getting 4 heads and 6 tails
120 ways of getting 3 heads and 7 tails
45 ways of getting 2 heads and 8 tails
10 ways of getting 1 head and 9 tails
1 way of getting 0 heads and 10 tails

252 ways out of 1024 therefore for 10 tosses being 5 heads and 5 tails or just under 1 in 4 chances (24.609 375%)

Read off the middle figure in the even rows to see that there are ,,,

70 ways out of 256 therefore for 8 tosses being 4 heads and 4 tails (27.34%)
20 ways out of 64 therefore for 6 tosses being 3 heads and 3 tails (31.25%)
6 ways out of 16 therefore for 4 tosses being 2 heads and 2 tails (37.5%)
2 ways out of 4 therefore for 2 tosses being 1 head and 1 tail (50%)

So for the person who thought it might be equal numbers of heads and tails 50% of the time, it is, but only when the number of tosses is two. Thereafter it starts to fall away, 37.5% for 4 tosses, 31% for 6 tosses, 27% for 8 tosses and 24% for 10 tosses ... this should not entirely surprise us, however, as the more tosses there are, the more possible outcomes there are (as many as 11 possible outcomes for 10 tosses but just the 3 possible outcomes for 2 tosses).

Consulting a larger version of the triangle in the first link we can see that

For 12 tosses 924 out of 4096 are 6 heads and 6 tails (22.55%)
For 14 tosses 3432 out of 16384 are 7 heads and 7 tails (20.95%)
For 16 tosses 12870 out of 65536 are 8 heads and 8 tails (19.64%)

Answer 2

Use a binomial distribution with n=10, x=5, and p=0.5. The probability is 0.24609.

Answer 3

10C5 = 10!/5!5! = 10*9*8*7*6*5*4*3*2/5*4*3*2*5*4*3*2
=7*6*4*3 = 504
10P10 = 10!/0! = 10*9*8*7*6*5*4*3*2 = 3628800
since the odds are 10C5/10P10 = 504/3628800 or 1/7200

Answer 4

* p= 1/2 f1= 5/10 f2=5/10

z=1/2 x 5 /10 = 1/2 x1 /2 = 1/4

A= 1/4+1/4+1/4+1/4 =1

Answer 5

(10C5)/2^10 = 63/256

Answer 6

24.61% I promise you THIS is the correct answer. I don't want to go into the details.

Answer 7

1/200.10(coins)x2(heads and tails)x10(different possibilities)-I guess.

Answer 8

intuition tells me 50% probability,...but I am not 100% certain....