Question

Two-headed coin Out of 100 coins one has heads on both sides. One coin is chosen at random and flipped
two times. What is the probability to get
(a) two heads?
(b) two tails?Two-headed coin Out of 100 coins one has heads on both sides. One coin is chosen at random and flipped
two times. What is the probability to get
(a) two heads?
(b) two tails?

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Answer 1

Answer:

(b) two tails?Two-headed coin Out of 100 coins one has heads on both sides. One coin is chosen at random and flipped

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Answer 2

Given:

• 1 coin out off 100 coins is chosen at random.
• 1 out of 100 coins has heads on both sides.
• One coin is chosen at random and flipped two times.

To Find:

The probability of getting

a) Two heads

b) Two tails

Solution:

First, we must understand the conditions posed in the question.

• A coin has two sides; heads and tails i.e the event of flipping a coin has only 2 outcomes (either we get a head or a tail)
• The probability of an event occurring is 1 always. Therefore, the probability of the two outcomes (head/tail) will be equal.

P(heads) =  $\frac{1}{2} = 0.5$

P(tails) = $\frac{1}{2} = 0.5$

Also, we must find that,

The probability of choosing the coin with heads on each side is $\frac{1}{100}$

P(two-headed coin) = $\frac{1}{100}$

Hence, the probability of choosing a normal coin will be

P(normal coin) = $1 - \frac{1}{100} = \frac{99}{100}$

Remember, the probability of all the outcomes of an event must always add up to 1.

Lastly, on the coin with two heads, the probability of getting heads will be 1 as the outcome of flipping the coin will undeniably be heads.

P(heads on two-headed coin) = 1

The probability of getting a tail on this coin is 0.

P( tails on two-headed coin) = 0

Recall,

• the term "and" suggests the multiplication of probabilities
• the term "or" suggests the addition of probabilities

a)

Probability of getting two heads⇒

Different ways by which we may obtain heads two times are:

1. Choosing a coin with heads on each side ⇔ obtaining heads two times on the two-headed coin (way 1)

                                                 OR

     2. Choosing a normal coin ⇔ obtaining heads two times (way 2)

P(way 1) ⇒

= P(choosing a two-headed coin) and P(obtaining heads on the two-headed coin) and P(obtaining heads on the two-headed coin)

= P(choosing a two-headed coin) * P(obtaining heads on the two-headed coin) *P(obtaining heads on the two-headed coin)

=$\frac{1}{100}$ * 1 * 1

= $\frac{1}{100} = 0.01$

  P(way 1) = $0.01$

P(way 2) ⇒

= P(Choosing a normal coin) and P(heads) and P(heads)

= P(Choosing a normal coin) * P(heads) * P(heads)\

= $\frac{99}{100} * 0.5 * 0.5$

= $0.2475$

  P(way 2) = $0.2475$

Finally,

P(two heads) = P(way 1) or P(way 2)

                      = P(way 1) + P(way 2)

                      = $0.01 + 0.2475$

                      = $0.2575$

Hence, the probability of getting two heads is $0.2575$.

b)

Probability of getting tails ⇒

We can obtain two tails consecutively by

(only one way to obtain two consecutive tails)

1. Choosing a normal coin and obtaining tails two times.

Now,

P(Probability of getting tails)

= P(normal coin) and P(tails) and P(tails)

= P(normal coin) * P(tails) * P(tails)

= $\frac{99}{100} * 0.5 * 0.5$

= $0.2475$

Hence, the probability of getting two heads is $0.2475$.