What are the odds your crush likes you? Bayes solved step by step

Problem. Start with an honest prior of P(L) = 0.10 that your crush likes you. If she likes you, she replies to your story 60% of the time and texts first 70% of the time; if she does not, she does so 30% and 20% of the time (she is just nice). Assume the two signs are independent given how she feels. What is the probability she likes you after both signs?

Step by step

1. Write Bayes' theorem: P(L|S) = P(S|L) × P(L) ÷ P(S), where L means she likes you and S is the sign you observed.

2. Start with the prior: P(L) = 0.10, so P(not L) = 0.90.

3. Sign 1 (she replied to your story): P(S) = 0.6 × 0.1 + 0.3 × 0.9 = 0.06 + 0.27 = 0.33.

4. Update after sign 1: P(L|S) = 0.06 ÷ 0.33 ≈ 0.18, so about 18%.

5. Sign 2 (she texted first) has likelihoods 0.7 if she likes you and 0.2 if not. Stack both signs from the original prior, with no rounding in between.

6. If she likes you: 0.1 × 0.6 × 0.7 = 0.042. If she does not: 0.9 × 0.3 × 0.2 = 0.054.

7. Normalize: P = 0.042 ÷ (0.042 + 0.054) = 0.042 ÷ 0.096 = 0.4375.

8. So two small signs moved the odds from 10% to about 44%.

Answer: After she replies to your story and texts first, the probability she likes you is 0.042 ÷ 0.096 = 0.4375, about 44% (10% → 18% → 43.75%).

The result depends entirely on the assumed starting 10% and the made-up reply rates, and it treats the two signs as independent, so use it as an illustration, not a measurement of a real person.