Question

two fair coins are tossed (say a dime and a quarter). the two possible outcomes for a single coin are heads, h, and tails, t. give each of the following.
(a) the sample space
(b) the probability of heads on the dime
(c) the probability of heads on the quarter
(d) the probability of getting both heads
(e) the probability of getting the same outcome on both coins
(a) the sample space is
(use a comma to separate answers as needed.)
(b) the probability of heads on the dime is
(type an integer or a simplified fraction.)
(c) the probability of heads on the quarter is
(type an integer or a simplified fraction.)
(d) the probability of getting both heads is
(type an integer or a simplified fraction.)
(e) the probability of getting the same outcome on both coins
(type an integer or a simplified fraction.)

Explanation:

Step1: Determine the sample space

When two fair coins (dime and quarter) are tossed, each coin has 2 outcomes (heads \( h \) or tails \( t \)). So the sample space is the set of all possible pairs: \( \{ (h,h), (h,t), (t,h), (t,t) \} \).

Step2: Probability of heads on the dime

The dime has 2 outcomes (\( h \) or \( t \)). The number of favorable outcomes (heads on dime) is 2 (when dime is \( h \), quarter can be \( h \) or \( t \))? Wait, no. Wait, for the probability of heads on the dime, we look at the dime's outcome. The dime has 2 possible outcomes: \( h \) or \( t \). The favorable outcome (heads) is 1? Wait, no. Wait, when we toss two coins, the sample space is 4 outcomes. For the dime's outcome: in the sample space, the dime is \( h \) in 2 cases: \( (h,h) \) and \( (h,t) \). So the number of favorable outcomes is 2, total outcomes is 4. So probability is \( \frac{2}{4} = \frac{1}{2} \).

Step3: Probability of heads on the quarter

Similar to the dime, the quarter is \( h \) in 2 cases: \( (h,h) \) and \( (t,h) \). So number of favorable outcomes is 2, total is 4. Probability is \( \frac{2}{4} = \frac{1}{2} \).

Step4: Probability of both heads

The favorable outcome is \( (h,h) \), which is 1 case. Total outcomes is 4. So probability is \( \frac{1}{4} \).

Step5: Probability of same outcome on both coins

The same outcome means both heads \( (h,h) \) or both tails \( (t,t) \). So number of favorable outcomes is 2. Total outcomes is 4. So probability is \( \frac{2}{4} = \frac{1}{2} \).

Answer:

s:
(a) The sample space is \( \{ (h,h), (h,t), (t,h), (t,t) \} \)
(b) The probability of heads on the dime is \( \frac{1}{2} \)
(c) The probability of heads on the quarter is \( \frac{1}{2} \)
(d) The probability of both heads is \( \frac{1}{4} \)
(e) The probability of same outcome is \( \frac{1}{2} \)