a game of chance involves spinning a wheel that has 4 equally likely areas to land on areas colored red, blue, green, and white. If jim plays the game 5 times, what is the probability he wins 3 or fewer times
To determine the probability that Jim wins 3 or fewer times out of 5 games, we need to calculate the probability for each outcome (winning or losing) and then sum them up.
The probability of winning in a single game is 1/4, as there are 4 equally likely areas on the wheel. Thus, the probability of losing in a single game is 1 - 1/4 = 3/4.
Now let's calculate the probability of Jim winning exactly 0 times (losing all 5 games):
Probability of losing in a single game: 3/4
Probability of losing all 5 games: (3/4)^5
Next, let's calculate the probability of Jim winning exactly 1 time:
Probability of winning in a single game: 1/4
Probability of losing in the other 4 games: (3/4)^4
Probability of winning 1 and losing 4 games: (1/4) * (3/4)^4
We can continue this pattern for winning 2 or winning 3 games and calculate their probabilities using the same logic.
Finally, to determine the probability of Jim winning 3 or fewer times, we sum up the probabilities of winning 0, 1, 2, and 3 games:
Probability of winning 0 games: (3/4)^5
Probability of winning 1 game: (1/4) * (3/4)^4
Probability of winning 2 games: (1/4)^2 * (3/4)^3
Probability of winning 3 games: (1/4)^3 * (3/4)^2
P(Jim wins 3 or fewer times) = Probability of winning 0 games + Probability of winning 1 game + Probability of winning 2 games + Probability of winning 3 games
Therefore, to find the answer, you need to calculate the four probabilities and sum them up.