Questions: If a gambler is betting on a coin flip and will win 10 if the coin lands on heads and win nothing if the coin lands on tails, what is the expected value of the coin toss?

Calculate the expected value of the coin toss.

Define the possible outcomes and their probabilities.

The coin has two possible outcomes: heads (probability \( P(H) = 0.5 \)) and tails (probability \( P(T) = 0.5 \)).

Define the payouts for each outcome.

If the coin lands on heads, the gambler wins \$10. If it lands on tails, the gambler wins \$0.

Calculate the expected value.

The expected value \( E \) is calculated as:
\[ E = (P(H) \times \text{Payout for heads}) + (P(T) \times \text{Payout for tails}) \]
Substituting the values:
\[ E = (0.5 \times 10) + (0.5 \times 0) = 5 + 0 = 5 \]

The expected value of the coin toss is \\(\boxed{5}\\).

The expected value of the coin toss is \\(\boxed{5}\\).