In John’s cattle pound, there are 5 bulls and 15 cows. In Jane’s cattle pound,
there are 2 bulls and 8 cows. John and Jane agree to exchange one animal
each. After John randomly selects one animal and sends it to Jane, Jane
randomly selects one animal and sends it to John.

(i) Draw the relevant sample space in a tree diagram.

(ii) Using it, find the probability of each of the following events.
     (a) There being a reduction in the number of bulls in John’s pound
         because of the exchange
     (b) There being an increase in the number of bulls in John’s pound
         because of the exchange
     (c) There being no difference in the number of bulls and cows in each of the
         two pounds because of the exchange

(iii) Now suppose they exchange animals in a way different to that mentioned
      above. Suppose John and Jane randomly take one animal from each of
      their pounds and go to their friend Adam’s house and exchange the cattle
      there and bring the exchanged cattle back to the pound. Find the probabilities
      of the events given in (ii) above for this random experiment. 

in Statistics Answers by Level 2 User (1.7k points)

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Tree diagrams added. I hope my revised calculations are correct.

PART A and PART B correspond to (iii) in the question; PART A involves John and Jane only and PART B involves John, Jane and Adam. 

PART A

John's probability of picking a bull from his cattle is 5/20=1/4 and 3/4 for a cow. After receiving one animal from John, Jane now has 11 animals in two scenarios (B=bull, C=cow):

Jane receives a bull: 3B+8C: Jane's probability of picking a bull from her cattle is 3/11 and 8/11 for a cow.

Jane receives a cow: 2B+9C: Jane's probability of picking a bull from her cattle is 2/11 and 9/11 for a cow.

(i) Loss and gain from John's point of view:

1. -B+B=0→(5B+15C,2B+8C), p=1/4 * 3/11=3/44=0.0682

2. -B+C(4B+16C,3B+7C), p=1/4 * 8/11=2/11=0.1818

3. -C+B(6B+14C,B+9C), p=3/4 * 2/11=3/22=0.1364

4. -C+C=0(5B+15C,2B+8C), p=3/4 * 9/11=27/44=0.6136

Note that 3/44+2/11+3/22+27/44=1.

These are the 4 leaves of the tree of probabilities. There are 4 possible outcomes.

On the tree diagram multiply the probability fractions encountered on the path to a particular leaf. This product is shown on the leaves at D, E, F and G. For example, if we take the path ACF, we are saying that John sends Jane a bull and Jane sends John a cow, probability p=1/4*8/11=2/11, leaf2 (point F in the diagram) in the sample space.

Sample space of animal configurations with ordered pairs (John's cattle,Jane's cattle):

{ (5B+15C,2B+8C) (4B+16C,3B+7C) (6B+14C,B+9C) (5B+15C,2B+8C) }

Corresponding sample space of probabilities:

{ 3/44 2/11 3/22 27/44 } 

(ii)

 (a) leaf2=2/11

 (b) leaf3=3/22

 (c) leaf1+leaf4=3/44+27/44=15/22

PART B

John's probability of picking a bull from his cattle is 5/20=1/4 and 3/4 for a cow as in PART A.

Jane's probability of picking a bull from her cattle is 2/10=1/5 and 4/5 for a cow. This differs from PART A.

(i) Loss and gain from John's point of view:

1. -B+B=0(5B+15C,2B+8C), p=1/4 * 1/5=1/20=0.05

2. -B+C(4B+16C,3B+7C), p=1/4 * 4/5=1/5=0.2

3. -C+B(6B+14C,B+9C), p=3/4 * 1/5=3/20=0.15

4. -C+C=0(5B+15C,2B+8C), p=3/4 * 4/5=3/5=0.6

Note that 1/20+1/5+3/20+3/5=1.

These are the 4 leaves of the tree of probabilities. There are 4 possible outcomes.

Sample space of cattle with ordered pairs as before:

{ (5B+15C,2B+8C) (4B+16C,3B+7C) (6B+14C,B+9C) (5B+15C,2B+8C) }

Corresponding sample space of probabilities:

{ 0.05 0.2 0.15 0.6 }

(ii)

 (a) leaf2=1/5=0.2

 (b) leaf3=3/20=0.15

 (c) leaf1+leaf4=1/20+3/5=13/20=0.65

by Top Rated User (1.1m points)
reshown by

Thank you so much Rod. Please can you say how to draw tree diagram for this.

Give me some time and I'll work one out. Probably some time tomorrow.
Great. Thank you

Related questions

1 answer
asked Sep 15, 2016 in Statistics Answers by math93 Level 2 User (1.7k points) | 729 views
1 answer
asked Oct 22, 2013 in Other Math Topics by anonymous | 516 views
1 answer
1 answer
asked Jul 13, 2013 in Statistics Answers by Leon89 Level 1 User (160 points) | 1.2k views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users