A 2n digit number starts with 2 and all its digits are prime, then find the probability that the sum of all 2 consecutive digits of the number is prime? The digits of the prime number are from {2,3,5,7} and the sum of consecutive digits will be prime if the number is something like 2325...but I cannot figure out how to satisfy the condition of the entire number being prime. Please help.
2026-03-28 22:06:22.1774735582
Problem involving probability and prime numbers.
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If I am inferring correctly your question, the digits will have to have the following pattern $$\underbrace{2,\,(5\text{ or }3)}_{\text{pair 1}},\underbrace{\,2,\,(5\text{ or }3)}_{\text{pair 2}},\dots,\,\underbrace{2,\,(5\text{ or }3)}_{\text{pair n}}.$$ I think this gives a probability of $\displaystyle \frac12\left(\frac18\right)^{n-1}$ assuming that the distribution on $\{2,3,5,7\}$ is uniform.