Question

. Female students at a certain university participate in sports with the following probabilities:

Soccer = 0.30 Basketball = 0.20

Tennis = 0.20

Both Soccer and Basketball = 0.10 Both Soccer and Tennis = 0.12 Both Basketball and Tennis = 0.08

All three Sports = 0.06

a. Construct the associated Venn diagram with ALL probabilities
specified.

You select one female student at random for an interview, determine the probability she plays

b. at least one of the three sports.

c. exactly one of the three sports.

d. exactly two of the three sports.

e. tennis given she does not play soccer.

f. tennis given she plays exactly one sport. g. exactly one sport
given she plays tennis.

Answer #1

let A,B and C are event of playing Soccer , Basketball and Tennis \

P(A)= | 0.3 | ||

P(B)= | 0.2 | ||

P(C)= | 0.2 | ||

P(AnB)=P(A)+P(B)-P(A U B)= | 0.1 | ||

P(BnC)=P(B)+P(C )-P(B U C)= | 0.08 | ||

P(AnC)=P(A)+P(C)-P(A U C)= | 0.12 |

P(AnBnC)=P(A)+P(B)+P(C)-P(AnB)-P(AnC)-P(BnC)+P(AuBuC)= | 0.06 |

a)

b)

least one=P(AUBUC)=P(A)+P(B)+P(C )-P(AnB)-P(BnC)-P(AnC)+P(AnBnC)= | 0.46 |

c)

exactly one =P(A)+P(B)+P(C )-2(P(AnB)+P(BnC)+P(AnC))+3*P(AnBnC) = | 0.28 |

d)

exactly two = | P(AnB)+P(AnC)+P(BnC)-3P(AnBnC)=0.12 |

e)

P(C|A^{c}) =P(Ac n C)/P(Ac)
=(0.2-0.12)/(1-0.3)=0.1143

f)

P(C|exactly one) =0.06/0.28 =0.2143

g)

P(exactly one |tennis) =0.06/0.2 =0.3

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