Answer:
Step-by-step explanation:Consider coin 1:
It has to be either HEADS or TAILS (obviously). If it's a “Fair” coin the probability of getting either a Heads or Tails is (1/2) for each option.
Conclusion(so far):
Prob (Heads Coin 1) = (1/2) AND
Prob (Tails Coin 1) = (1/2)
To meet the requirements of the question, the second coin toss must have the same result as for the first coin toss.
So if the result for Coin 1 was Heads then the result for Coin 2 must also be Heads.
So
(Prob (Heads Coin 1 AND Heads Coin 2) = (Prob Heads Coin1) * (Prob Heads Coin 2) = (1/2)*(1/2) = (1/4)
Conclusion (so far) (2 coins tossed):
Prob of two coin tosses BOTH giving Heads is (1/2)*(1/2) = (1/4)
AND similarly
Prob of two coin tosses BOTH giving Tails is also (1/2)*(1/2) = (1/4)
To meet the requirements of the question, the third coin toss will, by definition be either HEADS or TAILS.
AND
Prob (Coin 3 Heads) = (1/2) AND
Prob (Coin 3 Tails) = (1/2)
The question is clear: We are asked to find the Probability that the (3 tosses) are the same.
That is, if the first toss (Coin 1) is Heads, then the results for Coin 2 and Coin 3 tosses must also be Heads.
Similarly if the the first toss (Coin 1) is Tails, then the results for Coin 2 and Coin 3 tosses must also be Tails.
So for the tosses for Coins 1, 2 and 3 to each be Heads, we have:
Prob (Coin 1 Heads, Coin 2 Heads, Coin 3 Heads) = (1/2)*(1/2)*(1/2) = (1/8)
AND
Prob (Coin 1 Tails, Coin 2 Tails, Coin 3 Tails) = (1/2)*(1/2)*(1/2) = (1/8)
There are NO OTHER NUMBER COMBINATIONS WHICH MEET THE REQUIREMENTS OF THE QUESTION other than 3 Heads AND 3 Tails.
It is important to note THAT THE SITUATIONS ABOVE (3 Heads AND 3 Tails) are BOTH VALID SOLUTIONS.
Therefore
The Probability for 3 Heads is (1/8)
AND
The Probability for 3 Tails is (1/8)
BUT BOTH THESE SOLUTIONS ARE VALID PROBABILITIES and BOTH meet the requirements of the question.
Therefore the overall probability is THE SUM OF BOTH PROBABILITIES, that is:
The probability that after 3 tosses, the tosses of each coin ALL give the same result is:
(1/8) + (1/8) = (2/8) = (1/4) or 0.25 or 25%
This means that after 3 coin tosses there is a 25% probability that we will get either 3 Heads or 3 Tails.
(A note of interest: It would be an error to ignore the fact that there are TWO valid number combinations which meet the requirements of the question, not one).
(In my analysis above I have identified ONLY the valid-number combinations and spent little time considering non-valid-number combination).
