Final answer:
To find the value of 'a', set up a system of equations using the expected value of the random variable X and the total probability sum. Solve the equations to find the probability 'a' and round to the first decimal place.
Step-by-step explanation:
The random variable X has a given probability distribution, with known probabilities for values 1, 3, 4, and 5. The unknown probability of X being 2 is represented as 'a'. Given the expected value E(X) = 3.0, we can set up an equation based on the expected value formula: E(X) = ∑(x × P(x)), where ∑ represents the sum over all possible values of X.
The equation with the given values will be: 3.0 = (1 × 0.1) + (2 × a) + (3 × b) + (4 × 0.1) + (5 × 0.2). Since the total probability must sum up to 1, and we already have the sum of the other probabilities, we can also set up a second equation: 1 = 0.1 + a + b + 0.1 + 0.2.
Solve the system of equations for 'a' and 'b'. First, find the value of 'b' from the second equation, and then substitute it into the first equation to find 'a'. Calculate 'a' and round it to the first decimal place.