Probability

Maybe it’s just late but I seem to be confusing some of the basics of probablility. For example if the event is the probability of a six appearing in 4 throws of a dice then shouldnt the outcome be (1/6)^4.

However if we use the laws of probability where the P(Event)=1-P(Not event) then the number got from 1-(5/6)^4 does not match with the first method.

It’s late in my defense but what am I missing here?

It should be 1-P(Not Event)^4.

(5/6)^4 assumes that NONE of the 4 rolls results in a 6.

you are correct only in the second case it will be (1 - 5/6)^4

Agree. So 1- (5/6)^4 includes the probability that at least 1 six is rolled.

The second one is correct.

I am wondering why it is differs from the first. The probability of a six rolling in one die is 1/6. So with 4 attempts the value of (1/6)^4 should give the same outcome as the second method but it is not.

I misunderstood thie question at first.

(1/6) gives the probability that only a six is rolled. So (1/6)^4 gives the prbabiity that out of 4 rolls, only sixes are rolled. This is incorrect.

You are only looking for the probability that out of 4 rolls, at least one six is is rolled. So the probaiblity that at least one six is rolled, is one minus the porbability that no sixes are rolled. The porbaiblity that no six is rolled on an given roll is 5/6. The probability that no sixes are rolled on 4 rolls is (5/6)^4.

Therefore, the probabiltiy that AT LEAST ONE six is rolled is 1-(5/6)^4.

1 – (5/6)^4 is not the probability of getting all four sixes. It’s the probability of getting at least one six.

The complement of getting all four sixes is getting zero sixes, or getting one six, or getting two sixes, or getting three sixes.

  • P(zero sixes) = (5/6)^4
  • P(one six) = 4C1 × (5/6)³(1/6)
  • P(two sixes) = 4C2 × (5/6)²(1/6)²
  • P(three six) = 4C3 × (5/6)(1/6)³

So the alternative way of writing the probability of getting four sixes is:

1 – [(5/6)^4 + 4C1 × (5/6)³(1/6) + 4C2 × (5/6)²(1/6)² + 4C3 × (5/6)(1/6)³]

I knew it was a simple brainfart.

Thanks.