Question

You pick a card at random put it back and then pick another card at random what is the probability of picking a number greater than 5 and then picking a 5 right and then picking a 5 write your answer as a percentage

Answer 1

You have four cards numbered 4, 5, 6, and 7.

Step 1

To calculate the probability of picking a card at random, and that this card has a number greater than 5, you have to divide the number of successes by the number of possible outcomes.

Successes: You want to pick a card with a number greater than 5, there are only two cards that meet this condition, the card numbered 6 and the card numbered 7, so for this scenario, there are 2 successes.

Total outcomes: The number of outcomes is equal to the total number of cards you can pick from, in this case, the total number of outcomes is 4.

Next, calculate the probability of picking a card with a number greater than 5:

`\begin{gathered} P(X>5)=\frac{nºsuccesses}{Total\text{ }outcomes} \\ P(X>5)=(2)/(4) \\ P(X>5)=(1)/(2)=0.5 \end{gathered}`

The probability of picking a card with a number greater than 5 is 0.5.

Step 2

Next, you put the card back and pick another one at random.

You have to calculate the probability that this time you will pick the card numbered 5.

To calculate this probability you have to divide the number of successes by the total number of outcomes.

Successes: there is only one card with the number 5, so the number of successes is 1.

Total outcomes: since the first card that was drawn was returned to the deck, the total number of outcomes is still 4.

Calculate the probability of drawing a 5:

`\begin{gathered} P(X=5)=\frac{nºsuccesses}{Total\text{ }outcomes} \\ P(X=5)=(1)/(5)=0.2 \end{gathered}`

The probability of drawing a 5 is 0.2.

Finally, the probability that you have to determine is to "draw a card with a number greater than 5 and then pick a 5"

The event described is the intersection of both events "drawing a card greater than 5" and "picking a 5". Since the first card was returned to the deck before drawing the second card, both events are independent, which means that the probability of their intersection is equal to the product of the individual probabilities of the events, so that:

`P(X>5\cap X=5)=P(X>5)*P(X=5)=0.5*0.2=0.1`

The probability is 0.1.

Multiply the result by 100 to express it as a percentage:

`0.1*100=10\%`

The probability of picking a number greater than 5 and then picking a 5 is 10%.