A player of a video game is confronted with a series of opponents and has an 80% probability of defeating each one. Success with any opponent is independent of previous encounters. Until defeated, the player continues to contest opponents.
The correct answer and explanation is:
The correct answer is that the probability the player defeats nn opponents in a row is 0.8n0.8^n.
In this scenario, each time the player faces an opponent, they have an 80% (or 0.8) chance of winning. Since the outcomes of each encounter are independent, the probability of winning multiple matches in a row is simply the product of the probabilities for each individual match.
For example, the probability of winning the first two opponents would be: 0.8×0.8=0.640.8 \times 0.8 = 0.64
If the player wants to know the probability of defeating 3 opponents, it would be: 0.8×0.8×0.8=0.5120.8 \times 0.8 \times 0.8 = 0.512
In general, for nn opponents, the probability of winning each match in sequence is 0.8n0.8^n. This product decreases as the number of matches increases, since with each additional match, the player’s chances of winning the entire series decrease.
The player’s probability of continuing in the game (i.e., not being defeated after each round) follows this pattern. For instance, if they get through the first round, the chance of winning the next round drops to 0.8, and if they continue winning, the probability of getting through nn rounds is 0.8n0.8^n.
This model is an example of a geometric distribution, where the probability of success on each trial remains constant, and the trials are independent. The overall likelihood of winning many matches in a row decreases exponentially as the number of matches increases due to the compounding nature of independent probabilities.