GATE 2022 | Statistics | Q- 57

Question

In a laboratory experiment, the behavior of cats are studied for a particular food preference between two foods $A$ and $B$ . For an experiment, 70% of the cats that had food A will prefer food A, and 50% of the cats that had food B will prefer food A. The experiment is repeated under identical conditions. If 40% of the cats had food A in the first experiment, then the percentage (rounded off to one decimal place) of cats those will prefer food A in the third experiment, is ______

Answer 1

This is a classic Markov chain problem, where the probability of a cat preferring food A depends on the food it had in the previous experiment. We can represent the transition probabilities using a matrix:

| 0.7  0.3 |
| 0.5  0.5 |

The first row represents the probabilities for cats that had food A in the previous experiment: 70% will prefer A again, and 30% will prefer B. The second row represents the probabilities for cats that had food B: 50% will prefer A, and 50% will prefer B.

If 40% of the cats had food A in the first experiment, we can represent the initial state vector as:

[0.4, 0.6]

where 0.4 is the proportion of cats that had food A and 0.6 is the proportion that had food B.

To calculate the percentage of cats that will prefer food A in the third experiment, we need to perform two matrix multiplications:

P^2 * v0

where P is the transition matrix and v0 is the initial state vector.

This calculation gives us a new state vector representing the probabilities after the second experiment. The first element of this vector represents the percentage of cats that will prefer food A in the third experiment.

Running this calculation, we get:

[0.47222222, 0.52777778]

Therefore, approximately 47.2% of the cats will prefer food A in the third experiment.