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8) Of the 25 that like football, 10 like hockey as well so only 2 like hockey but not football, and 15 like football but not hockey. So we have 15+10+2=27 students out of 40 that like football or hockey or both. Also, 15+2=17 like football or hockey, but not both, so the probability (a) of selecting a student who likes one or the other (but not both) is 17/40=0.425 or 42.5%.

(b) 50-27=23 students like neither game, so that's 23/40=0.575 or 57.5% (this is 1 or 100%-(a)).

(c) Venn diagram:

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by Top Rated User (1.1m points)

9.

Tree diagram above.

Probability of all failing is 0.056.

Probability of at least one pass is the complement of this = 1-0.056=0.944.

The 8 outcomes (top to bottom), D=Dave, M=Mike, K=Ken, P=pass, F= fail:

DP,MP,KP (All pass=0.144)

DP,MP,KF

DP,MF,KP

DP,MF,KF

DF,MP,KP

DF,MP,KF (Only Mike passes=0.084)

DF,MF,KP

DF,MF,KF (All fail=0.056)

To find the probability, multiply together the probabilities encountered on the path to the "leaf". You can then fill in the gaps for the outcomes,

9) 

  1 2 3 4 5 6
HEAD H1 H2 H3 H4 H5 H6
TAIL T1 T2 T3 T4 T5 T6

6 of these possible outcomes contain a head, but we have to remove H3 because this event is head AND 3 and we want head OR 3. So there 6 outcomes: probability is 6/12=1/2 (H1 H2 H4 H5 H6 T3).

C1) There are 18 fruits in all.

a) 0, impossibility because there are no durians.

b) 8/18=4/9, because there are 8 apples.

c) 1-4/9=5/9, because this is the complement of (b).

C2)

The complete circle on the left represents P(A) and the one on the right P(B). X, Y and Z are regions such that:

P(A)=X+Y=0.5; P(A^B)=Y=0.35; P(AvB)=X+Y+Z=0.75.

Since Y=0.35, X=0.5-0.35=0.15 and Z=0.25

So P(B)=Y+Z=0.35+0.25=0.6.

5.

a) 6 outcomes out of 15 possible outcomes result in a green marble.

b) 5 outcomes out of 15 possible outcomes result in a blue marble.

6.

a) 1/6

b) 3/6=1/2 because there are 3 odd numbers: 1, 3, 5

7.

a) 3/6=1/2 because there are 3 primes: 11 13 17

b) also 1/2 because 15 21 51 are divisible by 3

8.

a) 12/57=4/19,

b) 25/57, because there 57 stickers in all

3.

a), c), d) 3/6=1/2 because there are 3 odd and 3 even numbers.

b) 4/6=2/3 because 1, 2, 3, 4 make 4 numbers less than 5.

e) An odd number is 1 3 5, a number less than 5 is 1 2 3 4. We have to eliminate elements that belong to both sets, so we end up with 2 4 5. Each one of these is EITHER an odd number OR less than 5. It isn't BOTH. So it satisfies the conditions. There are 3 numbers out of a possible 6, so the probability is 3/6=1/2 again.

4.

This diagram has been copied from 8c. The regions have been renamed.

Let's see how the figures were obtained. Region abbreviations: H=married male (husband), W=married female (wife), S=unmarried female (spinster), B=unmarried male (bachelor).

B+H=Male=381; H=168, so B=381-168=213.

Married=H+W=299, so W=299-168=131.

(Incidentally, S+B+H+W=655 people with multiple jobs, so S=143.)

We must only have unmarried males or married females to satisfy the conditions, so that's B+W=213+131=344. Therefore 344 people are EITHER married OR male, but NOT BOTH. That means we count bachelors and wives, and exclude husbands and spinsters. The probability of selecting a male or married person is therefore 344/655=0.5252 approx.

5) Use the diagram in C2. From the given figures we have 4 pieces of information:

  1. The probability of getting grade D only (implies a pass)=1/6
  2. The probability of not getting grade D (implies fail, or passing with a grade better than D)=5/6
  3. The probability of failing=1/9
  4. The probability of passing=8/9

Now let's define the regions X, Y, Z.

X probability of failing=1/9 (assuming that any grade below D is a fail; see (3) above)

Y probability of getting grade D=1/6 (see (1) above)

Z probability of getting better than grade D (we have to find)

X+Y=2/18+3/18=5/18, left circle (probability of failing or getting grade D)

Y+Z=probability of passing=8/9, right circle (see (4) above) so Z=8/9-1/6=13/18

X+Y+Z must equal 1 because it is certain that a student will pass or fail. And 1/9+1/6+13/18 does add up to 1. Also, (2) above is X+Z=1/9+13/18=15/18=5/6, so this fits, too.

Probability of getting a better grade than D is Z=13/18.

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