Consider the 6 x 6 square in the figure. Let A1, A2, ........, A49 be the points of intersections (dots in the picture) in some order. We say that Ai and Aj are friends if they are adjacent along a row or a column. Assume that each point Ai has an equal chance of being chosen. Let i p be the probability that a randomly chosen point has i many friends, i = 0,1,2,3,4. Let X be a random variable such that for i = 0,1,2,3,4, the probability P(X = i) =pi. Then the value of 7E(X) is
Correct Answer: 24
Solution and Explanation
Calculation of Probability of Points Having Friends
We are given a grid where each point can have 0, 1, 2, 3, or 4 friends. The following is provided:
- Number of points having 0 friends = 0
- Number of points having 1 friend = 0
- Number of points having 2 friends = 4
- Number of points having 3 friends = 5 × 4 = 20
- Number of points having 4 friends = 49 - 24 = 25
Step 1: Probability Calculation
The probability distribution for the points having 0, 1, 2, 3, or 4 friends is given by:
\(p_i = \frac{0}{49}, \frac{4}{49}, \frac{20}{49}, \frac{25}{49}\)
Step 2: Calculation of Expected Value
The expected value \( E(X) \) is calculated using the formula:
\(E(X) = \sum_{i=0}^{4} p_i X_i\)
Where \( X_i \) represents the number of friends, and \( p_i \) represents the corresponding probabilities. Substituting the values into the formula:
\(E(X) = 7 \left( 0 \times 0 + \frac{4}{49} \times 2 + \frac{20}{49} \times 3 + \frac{25}{49} \times 4 \right)\)
Simplifying:
\(E(X) = 7 \left( 0 + \frac{8}{49} + \frac{60}{49} + \frac{100}{49} \right)\)
\(E(X) = 7 \times \frac{168}{49} = \frac{1176}{49} = 24\)
Final Answer
The expected value \( E(X) \) is 24.
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.