If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest

[BTGModeratorVI]

If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

Answer: E
Source Veritas Prep

[Brent@GMATPrepNow]

If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

Answer: E
Source Veritas Prep

When it comes to probability questions involving "at least," it's best to try using the complement.

That is, P(Event A happening) = 1 - P(Event A not happening)
So, here we get: P(getting at least one heads and at least one tails) = 1 - P(not getting at least one heads and at least one tails)

What does it mean to not get at least one heads and at least one tails? It means getting EITHER zero heads OR zero tails.

So, we can write: P(getting at least one heads and at least one tails) = 1 - P(getting zero heads OR zero tails)
= 1 - P(getting ALL tails OR ALL heads)

P(getting ALL tails OR ALL heads)
= (tails on 1st toss AND tails on 2nd toss AND tails on 3rd toss AND tails on 4th toss AND tails on 5th toss AND tails on 6th toss OR heads on 1st toss AND heads on 2nd toss AND heads on 3rd toss AND heads on 4th toss AND heads on 5th toss AND heads on 6th toss)
= [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2] + [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2]
= [1/64] + [1/64]
= 2/64
= 1/32

So, P(getting at least one heads and at least one tails) = 1 - 1/32
= 31/32
≈ 97%

Answer: E

Cheers,
Brent

[Scott@TargetTestPrep]

If six coins are flipped simultaneously, the probability of getting at least one heads and at least one tails is closest to

A) 3%
B) 6%
C) 75%
D) 94%
E) 97%

Answer: E

Solution:

There are 2^6 = 64 ways the six coins could be flipped. Of the these 64 ways, only two of them (all heads and all tails) do not have at least one head or at least one tail. Therefore, the probability of getting at least one head and at least one tail is 62/64 = 0.96875 ≈ 97%.

Answer: E