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Senior Manager  B
Joined: 17 Sep 2013
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Concentration: Strategy, General Management
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What is the probability of getting exactly three heads on  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 74% (01:21) correct 26% (01:20) wrong based on 716 sessions

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What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

I am looking for a method better than counting...Consider if the Q had 10 coin flips or so..i.e counting would not be a feasible or practical option

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Re: What is the probability of getting exactly three heads on  [#permalink]

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JusTLucK04 wrote:
What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

I am looking for a method better than counting...Consider if the Q had 10 coin flips or so..i.e counting would not be a feasible or practical option

$$P(HHHTT)=\frac{5!}{3!2!}*(\frac{1}{2})^3*(\frac{1}{2})^2=\frac{10}{32}=\frac{5}{16}$$, we need to multiply by $$\frac{5!}{3!2!}$$ because HHHTT outcome can occur in several ways: HHHTTT, HHTHT, HTHHT, ..., TTHHH ($$\frac{5!}{3!2!}$$ is permutation of 5 letters HHHTT).

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Re: What is the probability of getting exactly three heads on  [#permalink]

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JusTLucK04 wrote:
What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

I am looking for a method better than counting...Consider if the Q had 10 coin flips or so..i.e counting would not be a feasible or practical option

i use this method, hope it helps
5!/3!*2!*1/2^5 = 5/16 (D)

5!/3!*2! = 5! as we flip the coin 5 times, divided by 3! as we want heads 3 times & 2! for tails

1/2^5 as probability of getting either a heads or a tails is 1/2 raise to 5 because we flip the coin 5 times

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Re: What is the probability of getting exactly three heads on  [#permalink]

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JusTLucK04 wrote:
What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

I am looking for a method better than counting...Consider if the Q had 10 coin flips or so..i.e counting would not be a feasible or practical option

5 Flips of a fair coin to get = HHHTT = no. of ways this can be achieved = 5!/3!x2! = 10

Probability to get any of the above 10 arrangements (HHHTT) = (1/2)^5 = 1/32

Total probability = 1/32 x 10 = 5/16
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Re: What is the probability of getting exactly three heads on  [#permalink]

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Total number of ways to get 3 heads and 2 tails ( p(h) =p(t)= 1/2 for each)
=2^5=32

# of ways 3 heads can be arranged in 5 tosses= 5c3=10
10/32=5/16 ANS
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Re: What is the probability of getting exactly three heads on  [#permalink]

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2 ways:
1/ 5C2 / 2^5 => there are totally 2^5 different results, but only 5C2 favor results,
1/ (1/8) * (1/4) * 5C2
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Re: What is the probability of getting exactly three heads on  [#permalink]

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HHHTT in any sequence would be required
Probability would be 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 5C3 = 1/32 x 10 = 5/16 (03 heads could be in any position, so 5C3)
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Re: What is the probability of getting exactly three heads on  [#permalink]

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JusTLucK04 wrote:
What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

I am looking for a method better than counting...Consider if the Q had 10 coin flips or so..i.e counting would not be a feasible or practical option

No. of ways of selecting 3 Heads = $$5C3$$
Total number = $$32$$
$$P = \frac{5C3}{32}$$
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Re: What is the probability of getting exactly three heads on  [#permalink]

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The answer is (5c2)*((1/2)^3)*((1/2)^2). Option D or 5/16 it is !!
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Re: What is the probability of getting exactly three heads on  [#permalink]

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JusTLucK04 wrote:
What is the probability of getting exactly three heads on five flips of a fair coin?

(A) 1/32
(B) 3/32
(C) 1/4
(D) 5/16
(E) 1/2

When a fair coin is flipped 5 times, there are 2^5 = 32 possible outcomes. Thus, each possible outcome is equally likely, with probability of 1/32.

The number of possible outcomes for getting 3 heads and 2 tails is 5!/(3! x 2!) = (5 x 4)/2 = 10.

Thus, the probability of getting 3 heads and 2 tails in 5 flips is (1/32) x 10 = 10/32 = 5/16.

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the probability of getting exactly three heads on  [#permalink]

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Hi All,

We're asked for the probability of getting EXACTLY three heads on five flips of a fair coin. This question can be approached in a couple of different ways, but they all involve a bit of 'Probability math.'

To start, since each coin has two possible outcomes, there are (2)(2)(2)(2)(2) = 32 possible outcomes from flipping 5 coins. To find the number of outcomes that are EXACTLY 3 heads, you can either use the Combination Formula or do some 'brute force' math and map out all of the possibilities.

By choosing 3 heads from 5 tosses, we can use the Combination Formula: N!/(K!)(N-K)! = 5!/(3!)(5-3)! = (5)(4)(3)(2)(1)/(3)(2)(1)(2)(1) = (5)(4)/(2)(1) = 10 possible ways to flip 3 heads from 5 tosses.

You could also list out the options:
HHHTT
HHTHT
HTHHT
THHHT

HHTTH
HTHTH
THHTH

HTTHH
THTHH

TTHHH

Either way, you have 10 total options that fit what we're looking for out of a total of 32 outcomes. 10/32 = 5/16

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_________________ Re: What is the probability of getting exactly three heads on   [#permalink] 08 Oct 2018, 22:09
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