Question 1100269: our labelled 2, 2, 3 and 5 are put in a hat, Three cards are drawn, one at a time, and the number showing is recorded, The eard is not replaced,
(a) Draw up the sample space using 2A and 2H distinguish between the two
b） How many outcomes are there in the sample space?
c）What is the probability that the number formed ignoring the A and a subserpts) is i）235 (ii) greater than 300 (iii) odd (iv) a multiple of 5 and is greater than 300?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! a) The possible sequences of cards are listed below,
distinguishing the ones labeled "2" by listing them as "2A" and "2H".
2A 2H 3
2A 2H 5
2A 3 2H
2A 3 5
2A 5 2H
2A 5 3
2H 2A 3
2H 2A 5
2H 3 2A
2H 3 5
2H 5 2A
2H 5 3
3 2A 2H
3 2A 5
3 2H 2A
3 2H 5
3 5 2A
3 5 2H
5 2A 2H
5 2A 3
5 2H 2A
5 2H 3
5 3 2A
5 3 2H
There is a total of 24 listed sequences, because there are
4 possibilities for the first card drawn,
3 for the second one,
and 2 for the third one,
for a total of  .
b) If we can tell the cards listed as 2A and 2H,
we would say there are  possible outcomes.
If we cannot tell them apart,
we would count  possible outcomes.
There are half as many sequences of numbers, as sequences of cards,
sequences of numbers,
because 2 will appear either once or twice in a sequence,
and the first 2 to appear is as likely to be 2A as 2H.
c) i) The probability that the number formed is 235 is  ,
because 235 is 1 of 12 possible sequences of numbers,
or because 2 of the 24 sequences of cards,
2A 3 5 and 2H 3 5 read as 235.
ii) For a sequence to read as greater than 300,
the first card drawn must be one of the  cards labeled 3 or 5,
which constitute  of the available cards,
so the probability that the number formed is greater than 300 is  .
iii) For a sequence to read as an odd number,
the last card must be odd.
All  cards are equally likely to be the last card drawn,
and exactly of them are labeled with an odd number (3 or 5),
so the probability that the number formed is odd is .
iv) As no card is labeled zero, the number formed will be a multiple of 5 if and only if the last card drawn is labeled 5.
For that to happen with a number greater than 300, the first card must be labeled 3.
That means that the sequence of numbers must be 325,
or in other words that the middle card be labeled 2.
That will happen in of the possible sequences of cards,
and the sequence 3 2 5 is  of the possible sequences of numbers.
As a consequence, the probability that the number formed
is a multiple of 5
and is greater than 300
is  .
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