Final answer:
The probability of each scenario in the raffle can be calculated by considering the total number of tickets and prizes. For the given question, the probability of all four organizers winning all the prizes is 0, the probability of exactly two organizers winning a prize is 0.0329, the probability of exactly one organizer winning a prize is 0.2245, and the probability of none of the organizers winning a prize is 0.3429.
Step-by-step explanation:
To calculate the probability of each scenario, we need to know the total number of tickets sold and the total number of prizes available. In this case, the total number of tickets sold is 50 and the total number of prizes is 3.
A. Probability of all four organizers winning all the prizes:
The probability of the first organizer winning a prize is 3/50. The probability of the second organizer winning a prize is 2/49, since there are now only 49 tickets left. Similarly, the probabilities for the third and fourth organizers are 1/48 and 0/47, respectively. Therefore, the probability of all four organizers winning all the prizes is (3/50) * (2/49) * (1/48) * (0/47) = 0.
B. Probability of exactly two organizers winning a prize:
The first organizer can win a prize in 3 different ways (first, second, or third prize). The second organizer can win a prize in 2 different ways, since the first prize is no longer available. The other two organizers will not win any prizes. Therefore, the probability of exactly two organizers winning a prize is (3/50) * (2/49) * (48/47) * (47/46) = 0.0329.
C. Probability of exactly one organizer winning a prize:
The first organizer can win a prize in 3 different ways. The second, third, and fourth organizers will not win any prizes. Therefore, the probability of exactly one organizer winning a prize is (3/50) * (47/49) * (46/48) * (45/47) = 0.2245.
D. Probability of none of the organizers winning a prize:
The probability of the first organizer not winning a prize is 47/50. The probability of the second organizer not winning a prize is 48/49, and so on. Therefore, the probability of none of the organizers winning a prize is (47/50) * (48/49) * (49/48) * (50/47) = 0.3429.