Final answer:
The probability that only one of Emily or Misha goes to work tomorrow, given their respective odds and assuming independent events, is calculated by converting the odds to probabilities and then determining the combined probability of either event occurring independently. The result is 78/121.
Step-by-step explanation:
The question is concerned with calculating the probability that only one of two people goes to work tomorrow, given the odds in favor of each person going. Emily's odds are provided as 8:3, and Misha's odds are provided as 2:9. Since these are independent events, we can calculate the probability of each event occurring and then find the probability of the combined event.
First, we convert the odds to probabilities. For Emily, the probability she goes to work is P(Emily goes) = 8 / (8+3) = 8/11. The probability she does not go to work is P(Emily doesn't go) = 3/11. For Misha, the probability she goes to work is P(Misha goes) = 2 / (2+9) = 2/11, and the probability she does not go to work is P(Misha doesn't go) = 9/11.
Next, we calculate the probability that only one of them goes to work, which can happen in two ways: Emily goes and Misha does not, or Emily does not go and Misha does. These probabilities are P(Emily goes) * P(Misha doesn't go) = (8/11) * (9/11), and P(Emily doesn't go) * P(Misha goes) = (3/11) * (2/11).
Adding these two probabilities together, we get the total probability that only one goes to work: (8/11) * (9/11) + (3/11) * (2/11). Calculating this, we get 72/121 + 6/121 = 78/121, which is the final answer.