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A biased die is used in a game - Leaving Cert Mathematics - Question 2 - 2011
Question 2
A biased die is used in a game. The probabilities of getting the six different numbers on the die are shown in the table below. | Number | 1 | 2 | 3 ... show full transcript
Worked Solution & Example Answer:A biased die is used in a game - Leaving Cert Mathematics - Question 2 - 2011
Step 1
Find the expected value of the random variable X
Answer
To determine the expected value of the random variable , we calculate it using the formula:
Substituting the values and probabilities from the table:
egin{align*}
E(X) &= (1 imes 0.25) + (2 imes 0.25) + (3 imes 0.15) + (4 imes 0.15) + (5 imes 0.10) + (6 imes 0.10)
&= 0.25 + 0.50 + 0.45 + 0.60 + 0.50 + 0.60
&= 2.90
\end{align*}
Thus, the expected value of is 2.90.
Step 2
Complete the sentence regarding the average winnings
Answer
To find the average winnings when using a fair die, we note that the probability of each number (1-6) is equal, specifically:
P( ext{number}) = rac{1}{6}
Then, the expected value with a fair die is:
E(X_{fair}) = rac{1 + 2 + 3 + 4 + 5 + 6}{6} = rac{21}{6} = 3.5
So, if you play the game many times with a fair die, you will win an average of €3.50 per game.
Now, considering the biased die, we previously calculated:
- Expected winnings with biased die: €2.90
- Cost to play: €3
Thus, if you play with the biased die, you will lose an average of:
In conclusion:
"If you play the game many times with a fair die, you will win an average of €3.50 per game, but if you play with the biased die you will lose an average of €0.10 per game."