Final answer:
To estimate probabilities for having a certain number of MBTI preferences in common, we use the independence and 50% chance assumption for each axis to calculate the binomial probabilities for 0, 1, 2, 3, or 4 preferences aligning, with the results rounded to four decimal places.
Step-by-step explanation:
To estimate the probability of having 0, 1, 2, 3, or 4 personality preferences in common, we would use the provided probabilities for each of the Myers-Briggs Type Indicator (MBTI) axes. These probabilities are for the individual preferences such as Extraversion (E), Introversion (I), Sensing (S), Intuition (N), Thinking (T), Feeling (F), Judging (J), and Perceiving (P). For simplicity, we assume each axis has a 50% chance for either preference and that they are uncorrelated.
If each axis is independent and has a 50% chance for each preference, then the probability of two individuals having exactly one preference in common would be calculated by looking at the combinations of the different ways they could match on one preference while differing on the others (which there are 4), then multiplying by the probability of a match (0.5) and the probability of not matching (also 0.5) three times. This gives us:
(4 choose 1) × (0.5)^1 × (0.5)^3 = 4 × 0.5 × 0.125 = 0.25 or 25
Similarly, the probability for zero preferences in common would involve all four preferences not matching, so it would be (0.5)^4, for two preferences, you would have (4 choose 2) × (0.5)^2 × (0.5)^2, and so on for three and four matches. Applying the binomial probability formula for each case:
0 preferences in common: (0.5)^4 = 0.0625 or 6.25%
1 preference in common: (4 choose 1) × (0.5)^4 = 0.25 or 25%
2 preferences in common: (4 choose 2) × (0.5)^4 = 0.375 or 37.5%
3 preferences in common: (4 choose 3) × (0.5)^4 = 0.25 or 25%
4 preferences in common: (4 choose 4) × (0.5)^4 = 0.0625 or 6.25%