Final answer:
The probability distribution f(x) of the number of emergency radios that work properly out of two randomly selected radios for a rescue mission is 1/6 for x=0, 3/4 for x=1, and 1/2 for x=2.
Step-by-step explanation:
The random variable X represents the number of emergency radios that work properly out of the two radios randomly selected for a rescue mission. Since there are four emergency radios available and only one of them does not work properly, the possible values for X are 0, 1, and 2.
To determine the probability distribution f(x) of X, we can use the concept of combinations. Let's calculate the probabilities for each possible value of X:
- X = 0: The probability that both radios selected are not working is the combination of selecting 2 radios from the 1 not working radio divided by the combination of selecting 2 radios from the 4 available radios. This can be calculated as C(1, 2) / C(4, 2) = 1/6.
- X = 1: The probability that one radio selected is working and one is not working is the combination of selecting 1 radio from the 3 working radios multiplied by the combination of selecting 1 radio from the 1 not working radio, divided by the combination of selecting 2 radios from the 4 available radios. This can be calculated as C(3, 1) * C(1, 1) / C(4, 2) = 3/4.
- X = 2: The probability that both radios selected are working is the combination of selecting 2 radios from the 3 working radios, divided by the combination of selecting 2 radios from the 4 available radios. This can be calculated as C(3, 2) / C(4, 2) = 3/6 = 1/2.