If you put high effort there is 30% chance of contributing $100 and 70% chance of $200, The cost of high effort to you is $20.

If you put low effort, there is 40% chance of contributing $20 and 60% chance of $100. The cost of low effort is zero.

If your contribution is $100 or higher, you will get a Bonus of B dollar. For what value of B, will you put high effort?

(Hint: you will put high effort if your expect income net of cost is higher from high effort than from low effort)

Question

If you put high effort there is 30% chance of contributing $100 and 70% chance of $200,  The cost of high effort to you is $20.

If you put low effort, there is 40% chance of contributing $20 and 60% chance of $100.  The cost of low effort is zero.

If your contribution is $100 or higher, you will get a Bonus of B dollar. For what value of B, will you put high effort?

(Hint: you will put high effort if your expect income net of cost is higher from high effort than from low effort)

BlackTom AI · Accepted Answer

We start by outlining the two choices and their monetary structures.
High effort: There is a 30% chance of contributing 100 and a 70% chance of contributing 200. The expected contribution is 0.3*100 + 0.7*200 = 30 + 140 = 170. The cost of high effort is 20, so the net from effort cost alone is 170 - 20 = 150. Since any contribution of 100 or more triggers a Bonus B, the total expected income with high effort is EV_high = 150 + B.
Low effort: There is a 40% chance of contributing 20 and a 60% chance of contributing 100. The expected contribution is 0.4*20 + 0.6*100 = 8 + 60 = 68. The cost of low effort is zero, so the net from effort cost is 68. The Bonus B applies only when the contribution is 100, which occurs with probability 0.6. Therefore, the expected value with low effort is EV_low = 68 + 0.6*B.
Now we compare the two options to determine when high effort yields a higher expected income: EV_high > EV_low translates to 150 + B > 68 + 0.6B.
Solving this inequality: subtract 0.6B from both sides to get 150 + 0.4B > 68. Subtract 68 from both sides to obtain 0.4B > -82. Divide by 0.4 to find B > -205.
This means that any Bonus B greater than -205 dollars makes high effort strictly better in expectation than low effort. Since B is typically nonnegative in these setups, any nonnegative B (including B = 0, B = 50, etc.) would satisfy the condition that you choose high effort.
To connect with the provided numerical answer: if B = 50, the inequality holds (150 + 50 = 200 on the high-effort side versus 68 + 0.6*50 = 98 on the low-effort side), reinforcing that high effort is preferable under this B value. The broader takeaway is that the threshold is B > -205, not a positive minimum like 50; the 50 value simply satisfies the condition.

COMM_V 295 105 106 2025W1 Participation/Math Quiz #3

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