describe joint probability for independent events and give an example of how you would calculate it. How is this different than conditional probability? The following payoff table gives the return on three alternative business decisions based on three possible scenarios – down, neutral or up. P(down), or the probability that the scenario is down = 0.30, P(neutral) = 0.50, and P(up) = 0.20. Based on this information, calculate the expected return for each alternative. What alternative would you recommend
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Joint probability refers to the probability of two or more independent events occurring simultaneously. In the context of independent events, the occurrence of one event does not affect the probability of the other event.
To calculate the joint probability of independent events, you multiply the probability of each event. Let's consider an example:
Suppose you have two events A and B, and the probabilities of these events occurring independently are P(A) = 0.4 and P(B) = 0.3. The joint probability of both events occurring simultaneously, denoted as P(A ∩ B), is calculated as follows:
P(A ∩ B) = P(A) * P(B) = 0.4 * 0.3 = 0.12.
In this example, the joint probability of events A and B occurring together is 0.12.
Now, let's discuss conditional probability. Conditional probability is the likelihood of an event occurring given that another event has already happened. In conditional probability, the occurrence of one event affects the probability of occurrence of another event.
To calculate conditional probability, you divide the joint probability by the probability of the given event. Mathematically, for events A and B, the conditional probability of event B given event A is denoted as P(B|A) and calculated as follows:
P(B|A) = P(A ∩ B) / P(A)
Conditional probability allows us to update our probabilities based on additional information that we have. It represents the probability of an event happening after we have received some prior information.
Now, let's move on to the second part of your question about expected returns and business decisions.
The expected return for each alternative can be calculated by multiplying the payoff of each scenario by its respective probability and summing them up.
Let's consider the given payoff table:
| | Down | Neutral | Up |
|------------|-------|----------|-------|
| Alt. 1 | $100 | $150 | $200 |
| Alt. 2 | $50 | $100 | $250 |
| Alt. 3 | $200 | $100 | $50 |
To calculate the expected return for each alternative, you multiply the payoff of each cell by its respective probability. Then, you sum up the products for each alternative.
For example, let's calculate the expected return for Alternative 1:
Expected Return for Alt. 1 = ($100 * P(down)) + ($150 * P(neutral)) + ($200 * P(up))
Substituting the given probabilities:
Expected Return for Alt. 1 = ($100 * 0.30) + ($150 * 0.50) + ($200 * 0.20) = $30 + $75 + $40 = $145.
Similarly, you can calculate the expected returns for Alternatives 2 and 3.
Finally, you would recommend the alternative with the highest expected return. In this case, you would recommend the alternative with the highest value calculated in the previous step.