Final answer:
The probabilities of the outcomes on the specially designed die are 1/5 each for outcomes 1, 2, 3, 4, and 5, and 0 for outcome 6. The probability of rolling an odd number (1, 3, or 5) is therefore 3/5.
Step-by-step explanation:
In a specially designed die where outcome 1 comes up a fifth of the time, outcome 6 never comes up, and the remaining outcomes (2, 3, 4, 5) are equally likely, we need to calculate the probabilities of all the possible outcomes.
The probability for outcome 1 is already given as 1/5. Since the die does not land on 6 at all, the probability for outcome 6 is 0. For the other outcomes (2, 3, 4, 5), we have a remaining probability of 4/5 (1 - 1/5), which needs to be equally distributed among the four numbers. Since there are four equally probable outcomes left, we divide 4/5 by 4, resulting in a probability of 1/5 for each one.
Therefore, the probabilities for the outcomes are:
- Probability of 1 = 1/5
- Probability of 2 = 1/5
- Probability of 3 = 1/5
- Probability of 4 = 1/5
- Probability of 5 = 1/5
- Probability of 6 = 0
For the probability of rolling an odd number (1, 3, or 5), we simply add their probabilities:
Probability of an odd number = Probability of 1 + Probability of 3 + Probability of 5
= 1/5 + 1/5 + 1/5 = 3/5