2. On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by \(X\).
- State two conditions needed for \(X\) to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\).
In the rest of this question you should assume that these conditions are satisfied.
- Find \(\mathrm { P } ( X \leqslant 6 )\).
In order to claim a free gift, 7 vouchers are needed.
- Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
- Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
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