6 A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).
- Sasha selects 10 beads at random, with replacement.
Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects.
Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
- Assume that \(p = 0.3\).
Find
- \(\mathrm { P } ( X \geqslant 5 )\),
- \(\operatorname { Var } ( X )\).
- In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\).
Estimate the value of \(p\).