An investment counselor calls with a hot stock tip. He believes that if the economy remains​ strong, the investment will result in a profit of ​$40,000. If the economy grows at a moderate​ pace, the investment will result in a profit of $20,000. However, if the economy goes into​ recession, the investment will result in a loss of ​$40,000. You contact an economist who believes there is a 20​% probability the economy will remain​ strong, a 70​% probability the economy will grow at a moderate​ pace, and a 10​% probability the economy will slip into recession. What is the expected profit from this​ investment?

The weather in Columbus is either good, indifferent, or bad on any given day. If the weather is good today, there is a 60% chance it will be good tomorrow, a 30% chance it will be indifferent, and a 10% chance it will be bad. If the weather is indifferent today, there is a 70% chance it will be good tomorrow, and a 20% chance it will be indifferent. Finally, if the weather is bad today, there is a 30% chance it will be good tomorrow and a 30% chance it will be indifferent. The stochastic matrix for this situation is shown to the right. In the long run, how likely is it for the weather in Columbus to be bad on a given day? P= 0.6 0.7 0.3 0.3 0.2 0.3 0.1 0.1 0.4 In the long run, how likely is it for the weather in Columbus to be bad on a given day? % (Round to the nearest integer as needed.)

The frequency of twin births has been increasing in recent years so that in 2022 about 4% of all births were twins. Twins may be either identical or fraternal. The probability of a birth being that of identical twins has remained fairly constant at 0.35%. What proportion of twin births is identical?

Joe loves going out to eat at area restaurants. Her favorite foods are Mexican, Italian, Chinese and Thai. On average, Joe pays $10.00 for a Mexican meal, $15.00 for an Italian meal, $9.00 for a Chinese meal, and $11.00 for a Thai meal. Joe's eating habits are predictable: There is a 70% chance that today's meal will be a repeat of yesterday's and an equal chance that he will switch to one of the remaining three. How often does Joe eat Mexican food?