GATE CS 2002 Q2.16 — Probability: Random Variables and Distributions

When four fair coins are tossed simultaneously, the total number of possible outcomes in the sample space is:

We want to find the probability of the event $E$: "at least one head and at least one tail turn up".

It is simpler to calculate the probability of the complementary event, $E^c$, which represents the outcomes where we do not get at least one head and one tail. This happens only when:

• All tosses result in Heads: $\{HHHH\}$ (1 outcome)
• All tosses result in Tails: $\{TTTT\}$ (1 outcome)

Thus, the number of outcomes in $E^c$ is $1 + 1 = 2$.

The probability of the complementary event is:

Therefore, the probability of the desired event $E$ is:

Hence, the correct option is C.