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Chapter 3: 79E (page 197)

Most likely coin-tossing sequence. In Parade Magazine鈥檚 (November 26, 2000) column 鈥淎sk Marilyn,鈥 the following question was posed: 鈥淚 have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.鈥

(1) H HHHHHHHHH

(2) H H T T H T T H HH

(3) T TTTTTTTTT

  1. Demonstrate that prior to actually tossing the coins, thethree sequences are equally likely to occur.
  2. Find the probability that the 10 coin tosses result in all heads or all tails.
  3. Find the probability that the 10 coin tosses result in a mix of heads and tails.
  4. Marilyn鈥檚 answer to the question posed was 鈥淭hough the chances of the three specific sequences occurring randomly are equal . . . it鈥檚 reasonable for us to choose sequence (2) as the most likely genuine result.鈥 If you know that only one of the three sequences actually occurred, explain why Marilyn鈥檚 answer is correct. [Hint: Compare the probabilities in parts b and c.]

Short Answer

Expert verified
  1. Every sequence has the same chance of succeeding is 0.000977
  2. The probability that the 10-coin tosses result in all heads or all tails is 0.001954.
  3. The probability that the 10-coin tosses result in a mix of heads and tails is 0.998.
  4. A sequence containing a combination of heads and tails is more likely to occurs than a sequence with all heads or all tails.

Step by step solution

01

Given information

Please choose one of the following three sequences as the outcome of the coin toss, I just completed. The only real sequence is one (and only one).

(1)HHHHHHHHHH

(2)HHTTHTTHHH

(3)TTTTTTTTTT

02

The three sequences are equally likely to occur

For an event A.

The require formula is PH鈭H鈭H鈭H鈭H鈭H鈭H鈭H鈭H鈭H.

Thus

P(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)=11024=0.000977

For an event B

The require formula isPH鈭H鈭T鈭T鈭H鈭T鈭T鈭H鈭H鈭H

Then

P(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)(0.50)=11024=0.000977

For event c.

The require formula isPT鈭T鈭T鈭T鈭T鈭T鈭T鈭T鈭T鈭T

Thus

.PT鈭T鈭T鈭T鈭T鈭T鈭T鈭T鈭T鈭T

Every coin toss offers 2 possible outcomes.Then the number of possible coin tossing

sequence for ten tosses is P(210)=1024.

Since, I have number of alternative coins tossing sequence from ten tosses. Therefore,

11024=0.000977

03

The probability that the 10-coin tosses result in all heads or all tails

The is probability is

P(A鈭C)=P(A)+P(C)=0.000977+0.000977=0.001954

04

Find the probability that the 10-coin tosses result in a mix of heads and tails

The probability is

P(headsandtailsaremixedtogether)=1-P(A鈭C)=1-0.001954=0.998

05

Find the result

Since, the probabilities of three specific sequence occurring randomly are similar, so it鈥檚 appropriate for everyone to choose sequence 2 as one of the most likely real results.

The probability that a coin will result in all tails or all heads is extremely small since there is only one sequence that result in A or B

Therefore, a sequence containing a combination of heads and tails is more likely to occurs than a sequence with all heads or all tails.

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