suppose a random variable X can equal to 1,2,3,4,or 5. if P(x<3)=.4 and p(x>3)=.5 (a)p(x=3) and b P(x<4)
a) P(x=3) = 0.1
b) P(x<4) = 0.9
To find the value of P(X=3), we can subtract the probability of X being less than 3 from the probability of X being greater than 3:
P(X=3) = P(X>3) - P(X<3)
Given that P(X<3) = 0.4 and P(X>3) = 0.5, we can substitute these values into the equation:
P(X=3) = 0.5 - 0.4
P(X=3) = 0.1
Therefore, P(X=3) is equal to 0.1.
To find P(X<4), we can use the information provided. Since X can take on the values 1, 2, 3, 4, or 5, we need to find the probabilities of X being less than 4 (1, 2, or 3):
P(X<4) = P(X=1) + P(X=2) + P(X=3)
Given that P(X<3) = 0.4 and P(X=3) = 0.1, we can substitute these values into the equation:
P(X<4) = P(X=1) + P(X=2) + P(X=3)
P(X<4) = 0.4 + 0.1
P(X<4) = 0.5
Therefore, P(X<4) is equal to 0.5.
To find the probability that X is equal to a specific value or falls within a specified range, we need to use the given information and apply the rules of probability.
a) To find P(X = 3), we can use the fact that the probabilities of all possible outcomes for a discrete random variable must sum to 1. In this case, since X can equal 1, 2, 3, 4, or 5, we have:
P(X < 3) + P(X = 3) + P(X > 3) = 1
Given that P(X < 3) = 0.4 and P(X > 3) = 0.5, we can substitute these values into the equation:
0.4 + P(X = 3) + 0.5 = 1
Simplifying the equation, we have:
P(X = 3) = 1 - 0.4 - 0.5
P(X = 3) = 0.1
Therefore, the probability that X is equal to 3 is 0.1.
b) To find P(X < 4), we can add up the probabilities of all the individual outcomes less than 4. Since X can equal 1, 2, 3, 4, or 5, we have:
P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3)
Given that P(X < 3) = 0.4 (which includes P(X = 1) and P(X = 2)), we can rewrite the equation:
P(X < 4) = P(X < 3) + P(X = 3)
P(X < 4) = 0.4 + 0.1
P(X < 4) = 0.5
Therefore, the probability that X is less than 4 is 0.5.