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The probability of Tom rolling a strike while bowling is 40% on any

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Bunuel
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The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   21 May 2019, 23:41
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55% (hard)

Question Stats:

62% (02:30) correct 38% (02:30) wrong based on 148 sessions
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The probability of Tom rolling a strike while bowling is 40% on any given frame. If Tom rolls 4 frames in a row, which of the following statements are true?

I. The probability of Tom rolling a strike on all 4 frames is greater than 3%.

II. The probability of Tom rolling no strikes in 4 frames is less than 10%.

III. Tom is equally likely to roll exactly 1 strike as to roll exactly 2 strikes in those 4 frames.


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III
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Official Answer
C
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The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   22 May 2019, 02:23
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I. The probability of Tom rolling a strike on all 4 frames is greater than 3%?
probability of 4 strikes = \(0.4^4 = 0.0256 < 0.03\) --> false

II. The probability of Tom rolling no strikes in 4 frames is less than 10%?
probability of no strikes = \(0.6^4 = 0.1296 > 0.1\) --> false

so A,B,D,E are eliminated & C is the answer

to confirm, we can check statement 3:

III. Tom is equally likely to roll exactly 1 strike as to roll exactly 2 strikes in those 4 frames? --> true
probability of exactly 1 = \(\frac{2}{5}*\frac{3}{5}*\frac{3}{5}*\frac{3}{5} * \frac{4!}{3!} = \frac{2*3*3*3*4}{5*5*5*5}\)
probability of exactly 2 = \(\frac{2}{5}*\frac{2}{5}*\frac{3}{5}*\frac{3}{5} * \frac{4!}{2!*2!} = \frac{2*3*3*3*4}{5*5*5*5}\)
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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   25 Jul 2020, 14:17
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Bunuel wrote:
The probability of Tom rolling a strike while bowling is 40% on any given frame. If Tom rolls 4 frames in a row, which of the following statements are true?

I. The probability of Tom rolling a strike on all 4 frames is greater than 3%.

II. The probability of Tom rolling no strikes in 4 frames is less than 10%.

III. Tom is equally likely to roll exactly 1 strike as to roll exactly 2 strikes in those 4 frames.


A. I only
B. II only
C. III only
D. I and III only
E. I, II and III


Solution:

Let’s analyze each Roman numeral statement.

I. Since the probability of Tom rolling a strike on all 4 frames is 0.4 x 0.4 x 0.4 x 0.4 = 0.0256 = 2.56%, we see that statement I is not true.

II. Since the probability of Tom rolling no strikes in 4 frames is 0.6 x 0.6 x 0.6 x 0.6 = 0.1296 = 12.96%, we see that statement II is not true.

At this point, we can see that choice C must be the correct answer since there is no choice that says “None.” However, let’s verify it anyway.

III. Since the probability of Tom rolling exactly 1 strike in 4 frames is 0.4 x 0.6 x 0.6 x 0.6 x 4C1 = 0.3456 and the probability of Tom rolling exactly 2 strikes in 4 frames is 0.4 x 0.4 x 0.6 x 0.6 x 4C2 = 0.3456, we see that statement III is true.

Answer: C
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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   27 Jul 2020, 00:59
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Solution


Given

    • The probability of Tom rolling a strike while bowling is 40% on any given frame
    • Tom rolls 4 frames in a row

To Find

    • The statements which are true.

Approach and Working Out


    • I. The probability of Tom rolling a strike on all 4 frames is greater than 3%.
      o The probability is given by,
      (\(\frac{40}{100}\))\(^4\)
      = 0.0256
      = 2.56%
      o This is false.

    • II. The probability of Tom rolling no strikes in 4 frames is less than 10%.
      o The probability is given by,
      (\(\frac{60}{100}\))\(^4\)
      = 0.1296
      = 12.96%
      o This is false.

    • III. Tom is equally likely to roll exactly 1 strike as to roll exactly 2 strikes in those 4 frames.
      o Exactly 1 strike = 4C1 × \(\frac{40}{100}\) × (\(\frac{60}{100}\))\(^3\) = 0.3456
      o Exactly 2 strike = 4C2 × (\(\frac{40}{100}\))\(^2\) × (\(\frac{60}{100}\))\(^2\) = 0.3456
      o This is true.


Correct Answer: Option C
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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   27 Jul 2020, 03:31
Why has everyone done 4C1 and 4C2 in the statement 3 solutions.?

Is it because 1 strike is selected out 4 chances and 2 is selected out of 4 chances or some other specific reason?

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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   27 Jul 2020, 03:58
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Nups1324 wrote:
Why has everyone done 4C1 and 4C2 in statement 3 solutions.?

Is it because 1 strike is selected out 4 chances and 2 is selected out of 4 chances or some other specific reason?

Posted from my mobile device


Hi Nups1324,

Yes, you are correct.
There are 4C1 and 4C2 ways to choose those strikes.

Regards!
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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   27 Jul 2020, 04:33
Oh! Alright.

Thank you so much for clearing my doubt.

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Re: The probability of Tom rolling a strike while bowling is 40% on any [#permalink] The probability of Tom rolling a strike while bowling is 40% on any   25 Apr 2024, 06:10
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