A coin is tossed 7 times. Find the probability of getting more heads than tails in all 7 tosses? A. 1/2

B. 63/128

C. 4/7

D. 61/256

E. 63/64

Assuming the coin is fair - P(H)=P(T)=1/2We can do as proposed by the explanation in your initial post: Total outcomes: 2^7

Favorable outcomes:

4 heads --> combination of HHHHTTT --> 7!/(4!*3!)=35 (# of permutation of 7 letters out of which 4 H's and 3 T's are identical);

5 heads --> combination of HHHHHTT --> 7!/(5!*2!)=21;

6 heads --> combination of HHHHHHT --> 7!/(6!*1!)=7;

7 heads --> combination of HHHHHHH --> 1;

P(H>T)=Favorable outcomes/Total outcomes=(35+21+7+1)/2^7=1/2.

BUT: there is MUCH simpler and elegant way to solve this question. Since the probability of getting either heads or tails is equal (1/2) and a tie in 7 (odd) tosses is not possible then the probability of getting more heads than tails = to the probability of getting more tails than heads = 1/2. How else? Does the probability favor any of tails or heads? (The distribution of the probabilities is symmetrical: P(H=7)=P(T=7), P(H=5)=P(T=5), ... also P(H>4)=P(T>4))

Answer: A.

If it were: A fair coin is tossed 8 times. Find the probability of getting more heads than tails in all 8 tosses?Now, almost the same here: as 8 is even then a tie is possible but again as distribution is symmetrical then

P(H>T)=\frac{1-P(H=T)}{2}=P(T>H) (so we just subtract the probability of a tie and then divide the given value by 2 as P(H>T)=P(H<T)). As

P(H=T)=\frac{8!}{4!*4!}=70 (# of permutation of 8 letters HHHHTTTT, out of which 4 H's and H T's are identical) then

P(H>T)=\frac{1-P(H=T)}{2}=\frac{1-\frac{70}{2^8}}{2}=\frac{93}{256}. You can check this in following way: total # of outcomes = 2^8=256, out of which in 70 cases there will be a tie, in 93 cases H>T and also in 93 cases T>H --> 70+93+93=256.

Hope it's clear.

Similar questions for practice:

probability-question-100222.html?hilit=coin%20tossed#p772756hard-probability-99478.html?hilit=coin%20tossedsome-ps-questions-need-explanation-99282.html?hilit=coin%20tossedprobability-question-gmatprep-85802.html?hilit=coin%20tossed _________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:

PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.

What are GMAT Club Tests?

25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership