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. Two distinct points are chosen randomly out of the points A1, A2, ........, A49. Let p be the probability that they are friends. Then the value of 7p is
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. Two distinct points are chosen randomly out of the points A1, A2, ........, A49. Let p be the probability that they are friends. Then the value of 7p is
Correct Answer: 0.5
Approach Solution - 1
Number of Ways to Select Adjacent Dots
Given: The number of ways to select 2 adjacent dots in a row is 6, and similarly, the number of ways to select 2 adjacent dots in a column is also 6.
Step 1: Total Ways of Selecting 2 Adjacent Dots from the Matrix
The total number of ways of selecting 2 adjacent dots from the matrix is:
\(6 \times 7 + 6 \times 7 = 84\)
Step 2: Calculating the Probability \( p \)
Now we can calculate the probability \( p \) of selecting 2 adjacent dots as:
\(p = \frac{84}{49} C_2 = \frac{84 \times 2}{49 \times 48}\)
Step 3: Multiplying by 7 to Get \( 7p \)
Next, we calculate \( 7p \):
\(7p = \frac{7 \times 84 \times 2}{49 \times 48} = \frac{1}{2}\)
Conclusion
Thus, the final value is:
0.50 is the correct answer.
Approach Solution -2
Selection of Adjacent Dots in a Matrix
Given: We are tasked with finding the probability \( \rho \) of selecting 2 adjacent dots from a matrix.
Step 1: Number of Ways to Select 2 Adjacent Dots
The number of ways to select 2 adjacent dots in a row is 6. Similarly, the number of ways to select 2 adjacent dots in a column is also 6.
The total number of ways to select 2 adjacent dots from the matrix is calculated as:
\(\text{Number of ways} = 6 \times 7 + 6 \times 7 = 84\)
Step 2: Calculating the Probability \( \rho \)
The probability \( \rho \) is calculated as:
\(\rho = \frac{84}{49} C_2 = \frac{84 \times 2}{49 \times 48}\)
Step 3: Final Calculation
Now, we multiply by 7 to get \( 7\rho \):
\(7\rho = \frac{7 \times 84 \times 2}{49 \times 48} = \frac{1}{2} = 0.50\)
Conclusion
The answer is: 0.50
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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.