Analyzing a Situation
Andrea rolls a number cube twice. She determines [tex]P( even, then odd )=\frac{1}{2}[/tex]. Which statements are accurate? Check all that apply.
A. Andrea's solution is incorrect.
B. The total number of possible outcomes is 12.
C. The probability of each simple event is [tex]\frac{1}{2}[/tex].
D. There are three possible outcomes on each number cube.
E. The probability of the compound event is less than the probability of either event occurring alone.
F. [tex]P( even, then odd )=P( odd, then even )[/tex]
Answer (1)
The accurate statements are: Andrea's solution is incorrect, The probability of each simple event is 2 1 , The probability of the compound event is less than the probability of either event occurring alone, P ( e v e n , t h e n o dd ) = P ( o dd , t h e n e v e n ) .
Explanation
Analyze the problem Let's analyze the situation. Andrea rolls a standard six-sided number cube twice and calculates the probability of rolling an even number followed by an odd number. We need to verify the accuracy of her calculation and the given statements.
Calculate individual probabilities First, let's calculate the probability of rolling an even number on a six-sided die. There are 3 even numbers (2, 4, 6) out of 6 possible outcomes (1, 2, 3, 4, 5, 6). So, the probability of rolling an even number is: P ( e v e n ) = 6 3 = 2 1 = 0.5 Similarly, the probability of rolling an odd number is: P ( o dd ) = 6 3 = 2 1 = 0.5
Calculate the combined probability Now, let's calculate the probability of rolling an even number, then an odd number. Since the two rolls are independent events, we multiply their probabilities: P ( e v e n , t h e n o dd ) = P ( e v e n ) × P ( o dd ) = 2 1 × 2 1 = 4 1 = 0.25 Andrea's solution states that P ( e v e n , t h e n o dd ) = 2 1 , which is incorrect.
Determine total possible outcomes The total number of possible outcomes when rolling a six-sided die twice is 6 × 6 = 36 .
Evaluate the statements Now, let's evaluate the given statements:
Andrea's solution is incorrect: This is accurate because we calculated the probability as 4 1 , not 2 1 .
The total number of possible outcomes is 12: This is incorrect . The total number of outcomes is 36.
The probability of each simple event is 2 1 : This is accurate . The probability of rolling an even number is 2 1 , and the probability of rolling an odd number is 2 1 .
There are three possible outcomes on each number cube: This is incorrect . There are six possible outcomes on each number cube (1, 2, 3, 4, 5, 6).
The probability of the compound event is less than the probability of either event occurring alone: This is accurate . The probability of the compound event (even, then odd) is 4 1 = 0.25 , which is less than the probability of either event occurring alone ( 2 1 = 0.5 ).
P ( e v e n , t h e n o dd ) = P ( o dd , t h e n e v e n ) : This is accurate . Since P ( e v e n ) = P ( o dd ) = 2 1 , then P ( e v e n , t h e n o dd ) = 2 1 × 2 1 = 4 1 and P ( o dd , t h e n e v e n ) = 2 1 × 2 1 = 4 1 .
Final Answer Therefore, the accurate statements are:
Andrea's solution is incorrect.
The probability of each simple event is 2 1 .
The probability of the compound event is less than the probability of either event occurring alone.
P ( e v e n , t h e n o dd ) = P ( o dd , t h e n e v e n )
Examples
Understanding probabilities helps in many real-life situations. For example, when playing board games that involve dice, knowing the probability of rolling certain numbers can inform your strategy. Similarly, in card games, understanding the likelihood of drawing specific cards can improve your decision-making. These concepts are also crucial in fields like finance, where assessing risk involves calculating probabilities of different outcomes. For instance, the probability of rolling an even number then an odd number is 4 1 , which means that, on average, this sequence will occur once every four attempts.
