Verified Answer :

  • Probability distribution tables list every possible value of a random variable alongside its associated probability.
  • Locate the target value \(X=3\) in the table.
  • Identify the value in the 'Probability: \(P(X)\)' column corresponding to \(X=3\).
  • The value is \(\boxed{0.2}\).

Explanation

  1. Analyze the probability distribution table

The problem asks us to determine the probability of a random variable \(X\) taking the value of \(3\), denoted as \(P(3)\), based on the provided probability distribution table. A probability distribution table displays the likelihood of each outcome for a random variable. The value in the 'Probability: \(P(X)\)' column corresponds directly to the probability that \(X\) takes the value listed in the adjacent 'X' column.

  1. Extract the value for P(3)

To find \(P(3)\), we simply look for the row where \(X = 3\) in the table. Scanning down the 'X' column:

  • For \(X = 0\), \(P(X) = 0.1\)
  • For \(X = 1\), \(P(X) = 0.2\)
  • For \(X = 2\), \(P(X) = 0.4\)
  • For \(X = 3\), \(P(X) = 0.2\)
  • For \(X = 4\), \(P(X) = 0.1\)

By identifying the row with \(X = 3\), we see that the corresponding probability is \(0.2\). Thus, \(P(3) = 0.2\).

Examples

Understanding probability distribution tables is essential in various fields, such as statistics, finance, and quality control. For example, if a store owner wants to know the probability of selling a specific number of units per day based on historical data, they can refer to a probability distribution table. This helps them manage inventory effectively by predicting demand trends. In this case, if \(X\) represents the number of daily sales, the table provides a quick way to assess the likelihood of different sales volumes.

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