Find P(X < 5) from the Given Discrete Probability Distribution

Understanding the Given Probability Distribution

This question is based on a discrete random variable and its probability distribution. The random variable X can take the values 0, 1, 2, 3, 4, and 5, while the corresponding probabilities are expressed in terms of an unknown constant k.

The given probability distribution is:

We need to calculate P(X < 5). However, before calculating this probability, we must determine the value of k. The value of k can be obtained using the fundamental property that the sum of all probabilities in a valid probability distribution must be equal to 1.

Basic Rule of a Discrete Probability Distribution

For any discrete random variable, the probabilities of all possible outcomes must add up to 1. Mathematically, this property is written as:

ΣP(X = x) = 1

This rule represents certainty. Since the random variable must take one of the listed possible values, the combined probability of all possible outcomes is 1.

Therefore, for the given distribution:

0 + k + 2k + 3k + 6k + 8k = 1

Solving this equation will give the value of k.

Step-by-Step Solution

Step 1: Add All the Probabilities

The probabilities corresponding to X = 0, 1, 2, 3, 4, and 5 are:

0, k, 2k, 3k, 6k, and 8k

According to the probability normalization rule:

0 + k + 2k + 3k + 6k + 8k = 1

Combining the terms containing k:

(1 + 2 + 3 + 6 + 8)k = 1

Adding the coefficients:

20k = 1

Therefore:

k = 1/20

Hence:

k = 0.05

Step 2: Interpret the Condition X < 5

The question asks us to calculate:

P(X < 5)

The symbol < means strictly less than. Therefore, we must include all possible values of X that are smaller than 5.

From the given distribution, these values are:

X = 0, 1, 2, 3, and 4

The value X = 5 must not be included because 5 is equal to 5 and is not less than 5.

Therefore:

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

Step 3: Substitute the Corresponding Probabilities

From the probability distribution:

P(X = 0) = 0

P(X = 1) = k

P(X = 2) = 2k

P(X = 3) = 3k

P(X = 4) = 6k

Therefore:

P(X < 5) = 0 + k + 2k + 3k + 6k

Combining the terms:

P(X < 5) = 12k

Step 4: Substitute the Value of k

We have already calculated:

k = 1/20

Therefore:

P(X < 5) = 12 × (1/20)

Thus:

P(X < 5) = 12/20

Simplifying the fraction:

P(X < 5) = 3/5

Converting the fraction into decimal form:

P(X < 5) = 0.6

Step 5: Round the Answer to One Decimal Place

The question asks for the answer rounded off to one decimal place. The calculated probability is exactly 0.6, which already contains one digit after the decimal point.

Therefore:

P(X < 5) = 0.6

Why X = 5 Is Not Included in the Calculation

The condition in the question is X < 5, which means that only values strictly smaller than 5 are included. Therefore, the probability corresponding to X = 5, which is 8k, must be excluded.

If the question had asked for P(X ≤ 5), then X = 5 would also have been included. In that case, every possible value of X would be included, and the total probability would be 1.

However, for the present condition:

X < 5 → X = 0, 1, 2, 3, 4

Therefore, the required sum is:

0 + k + 2k + 3k + 6k = 12k = 0.6

Alternative Solution Using the Complement Rule

The answer can also be calculated more quickly using the complement rule of probability. Since the only possible value that is not less than 5 is X = 5, we can write:

P(X < 5) = 1 − P(X = 5)

From the probability distribution:

P(X = 5) = 8k

Since k = 1/20:

P(X = 5) = 8 × (1/20)

Therefore:

P(X = 5) = 8/20 = 0.4

Using the complement rule:

P(X < 5) = 1 − 0.4

Hence:

P(X < 5) = 0.6

This alternative method confirms the result obtained by directly adding the probabilities of X = 0, 1, 2, 3, and 4.

Understanding the Role of the Constant k

The constant k ensures that the given values form a valid probability distribution. Without determining k, the individual probabilities cannot be expressed numerically.

After finding k = 0.05, the complete probability distribution becomes:

Adding all these probabilities gives:

0 + 0.05 + 0.10 + 0.15 + 0.30 + 0.40 = 1

This confirms that the calculated value of k produces a valid probability distribution.

Final Answer

The value of P(X < 5) is 0.6 when rounded off to one decimal place.