The probability of one random event occurring is the same as the product of the same events occurring together?

Answer 1

Consider a coin toss. The probability of the coin coming up head is 1/2, the probability of the coin coming up tails is 1/2. No matter how many times you flip the coin the probability of any particular toss coming up heads or tails is always 1/2.

BUT

If you consider two coins tosses as a set then you have four possible outcomes:

first coin heads - second coin heads, first coin heads - second coin tails,
first coin tails - second coin heads, first coin tails - second coin tails

for ease of demonstration lets denote heads as H and tails as T

the above combination would then be HH,HT,TH,TT

the probability of any one of the combinations would be 1/2 for the first coin and 1/2 for the second coin so all combinations have a probability of 1/4. (1/2x1/2=1/4 I hope you realize.)

But, if we don't care which coin is H or T then the two combinations HT and TH are the same and we combined their probabilities into one 1/4+1/4=1/2

HH,(HT,TH),TT

1/4 1/2 1/4