Most likely value (mode)

6 The probability that Sue completes a Sudoku puzzle correctly is 0.75 .

1. Sue attempts \(n\) Sudoku puzzles. Find the least value of \(n\) for which the probability that she completes all \(n\) puzzles correctly is less than 0.06 . Sue attempts 14 Sudoku puzzles every month. The number that she completes successfully is denoted by \(X\).
2. Find the value of \(X\) that has the highest probability. You may assume that this value is one of the two values closest to the mean of \(X\).
3. Find the probability that in exactly 3 of the next 5 months Sue completes more than 11 Sudoku puzzles correctly.

2．The discrete random variable \(X\) follows the binomial distribution $$X \sim \mathrm {~B} ( n , p )$$ where \(0 < p < 1\) ．The mode of \(X\) is \(m\) ．
（a）Write down，in terms of \(m , n\) and \(p\) ，an expression for \(\mathrm { P } ( X = m )\)
（b）Determine，in terms of \(n\) and \(p\) ，an interval of width 1 ，in which \(m\) lies．
（c）Find a value of \(n\) where \(n > 100\) ，and a value of \(p\) where \(p < 0.2\) ，for which \(X\) has two modes． For your chosen values of \(n\) and \(p\) ，state these two modes．