1 On any day, Kino travels to school by bus, by car or on foot with probabilities 0.2, 0.1 and 0.7 respectively. The probability that he is late when he travels by bus is \(x\). The probability that he is late when he travels by car is \(2 x\) and the probability that he is late when he travels on foot is 0.25 .
The probability that, on a randomly chosen day, Kino is late is 0.235 .
Find the value of \(x\).
Find the probability that, on a randomly chosen day, Kino travels to school by car given that he is not late.
3 Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.
Find the probability that a score of 17 or more is first obtained on the 6th throw.
Find the probability that a score of 17 or more is obtained in fewer than 8 throws.
4 The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.
Time taken \(( t\) minutes \()\)
\(0 \leqslant t < 20\)
\(20 \leqslant t < 40\)
\(40 \leqslant t < 50\)
\(50 \leqslant t < 60\)
\(60 \leqslant t < 100\)
Frequency
32
46
96
52
24
Draw a histogram to represent this information.
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From the data, the estimate of the mean time taken by these 250 employees is 43.2 minutes.
Calculate an estimate for the standard deviation of these times.
5 Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac { 1 } { 4 }\), independently of all other throws. Eric throws all three coins at the same time.
Events \(A\) and \(B\) are defined as follows.
\(A\) : all three coins show the same result
\(B\) : at least one of the biased coins shows a head
Show that \(\mathrm { P } ( B ) = \frac { 7 } { 16 }\).
Find \(\mathrm { P } ( A \mid B )\).
The random variable \(X\) is the number of heads obtained when Eric throws the three coins.
Draw up the probability distribution table for \(X\).
Find the number of different arrangements of the 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.
The 9 letters in the word ALLIGATOR are arranged in a random order.
Find the probability that the two Ls are together and there are exactly 6 letters between the two As.
Find the number of different selections of 5 letters from the 9 letters in the word ALLIGATOR which contain at least one A and at most one L.
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