Determine The Required Value Of The Missing Probability To Make The Distribution A Discrete Probability Distribution
A X 3 4 5 6 P(X) 0.34 ? 0.23 0.18
P(4)-(Type An Integer Or A Decimal) Ty
The Correct Answer and Explanation is :
To determine the missing probability ( P(4) ), we need to ensure that the given probability distribution satisfies the condition of a discrete probability distribution.
Step 1: Understanding a Discrete Probability Distribution
A discrete probability distribution must satisfy two key conditions:
- Each probability must be between 0 and 1: That is, ( 0 \leq P(X) \leq 1 ) for all values of ( X ).
- The sum of all probabilities must equal 1: That is,
[
P(3) + P(4) + P(5) + P(6) = 1
]
Step 2: Setting Up the Equation
From the given data:
- ( P(3) = 0.34 )
- ( P(4) = ? )
- ( P(5) = 0.23 )
- ( P(6) = 0.18 )
Using the probability sum rule:
[
0.34 + P(4) + 0.23 + 0.18 = 1
]
Step 3: Solving for ( P(4) )
First, sum up the known probabilities:
[
0.34 + 0.23 + 0.18 = 0.75
]
Now, solve for ( P(4) ):
[
P(4) = 1 – 0.75
]
[
P(4) = 0.25
]
Final Answer: ( P(4) = 0.25 )
Explanation (300 Words)
A probability distribution represents how probabilities are assigned to discrete outcomes of a random variable. In this case, the random variable ( X ) can take values 3, 4, 5, or 6, and each value has a corresponding probability ( P(X) ).
To ensure that this is a valid probability distribution, the sum of all probabilities must equal 1. Given that ( P(3) = 0.34 ), ( P(5) = 0.23 ), and ( P(6) = 0.18 ), we calculated the sum of these known probabilities as:
[
0.34 + 0.23 + 0.18 = 0.75
]
Since the total probability must be 1, the missing probability ( P(4) ) is found by subtracting this sum from 1:
[
P(4) = 1 – 0.75 = 0.25
]
Thus, ( P(4) = 0.25 ), making the probability distribution complete and valid.
This process ensures that the given data meets the conditions of a discrete probability distribution: all probabilities are within the range [0,1], and their total sum is exactly 1.
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