Let S={1,2,3,4,5,6}. Then the probability that a randomly chosen onto function G from S to S satisfies g(3)=2g(1) is?

Odds In Favor And Odds Against An Event In Probability

Both odds in favor and odds against provide different perspectives on the probability of an event and can be useful for comparing probabilities in different contexts.

• Odds in Favor: It’s a way of expressing the probability of an event occurring. It represents the ratio of favorable outcomes to unfavorable outcomes.

For example: if the odds in favor of winning a game are 2:1, it means there are 2 chances of winning for every 1 chance of losing.

• Odds Against: It’s the opposite of odds in favor. It expresses the probability of an event not occurring. It represents the ratio of unfavorable outcomes to favorable outcomes.

For example: if the odds against winning a game are 3:1, it means there are 3 chances of losing for every 1 chance of winning.

Solution

Step 1: Determine the favorable outcomes

To satisfy g(3)=2g(1),

we need to determine the mappings for g(1) and g(3).

Since,

g(1) can be mapped to any element in S except 2,

there are 5 choices for g(1).

Once g(1) is chosen,

g(3) must be twice the value of g(1).

Therefore,

there is only one valid choice for g(3) corresponding to each choice of g(1).

Thus, there are 5 favorable outcomes.

Step 2: Determine the total number of outcomes

The total number of onto functions from S to S can be calculated using the concept of permutations.

Since both the domain and codomain have 6 elements, there are 6! possible functions in total.

Therefore, the total number of outcomes is 6!

Step 3: Calculate the probability

The probability is given by the number of favorable outcomes divided by the total number of outcomes.

Probability = Number of favorable outcomes / Total number of outcomes

Probability = 5 / (6!)

= 5 / (6 × 5 × 4 × 3 × 2 × 1)

= 5 / 720

= 1 / 10

Therefore,

The probability that a randomly chosen onto function G from S to S satisfies g(3)=2g(1) is 1/10.

Summary

Let S={1,2,3,4,5,6}. Then the probability that a randomly chosen onto function G from S to S satisfies g(3)=2g(1) is?

Let S = {1,2,3,4,5,6}. Then the probability that a randomly chosen onto function G from S to S satisfies g(3)=2g(1) is 1/10.

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