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Chapter 7: Q17SE (page 496)

Question: There are three cards in a box. Both sides of one card are black, both sides of one card are red, and the third card has one black side and one red side. We pick a card at random and observe only one side.

a) If the side is black, what is the probability that the other side is also black?

b) What is the probability that the opposite side is the same color as the one we observed?

Short Answer

Expert verified

Answer

a) The probability of the other side of the card is black, if one side is black is \(\frac{2}{3}\).

b) The probability of opposite side is same color,as we observe the one side of the card \(\frac{2}{3}\).

Step by step solution

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01

Given data

In a box three cards in a box, both sides of one card is black, sides of one card is red, third card has one side black and one side red. We can pick a card randomly and observe only one side

02

Concept of Probability

Probability is simply how likely something is to happen. Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

Formula:

The probability \( = \frac{{{\rm{ favorable outcomes }}}}{{{\rm{ total outcomes }}}}\)

03

Calculation for the probability of the other side of the card is black 

a)

Here,\({B^\prime }\)indicates black side.

The possibilities are \(B{B^\prime }\;,\;{B^\prime }B\;,\;BR\;,\;RB\;,\;{R^\prime }R\;,\;R{R^\prime }\).

When it picks a card, the possibilities cards face is black are \(3\).

The possibilities cards when the other side is also black are \(2\).

Therefore, the required probability is \(\frac{2}{3}\).

04

Calculation for the probability that the opposite side is the same color

b)

Here, \({B^\prime }\) indicates black side.

The possibilities are \(B{B^\prime }\;,\;{B^\prime }B\;,\;BR\;,\;RB\;,\;{R^\prime }R\;,\;R{R^\prime }\).

When it picks a card, the total possibilities \(3\).

The possibilities cards when the opposite side is also same color are \(2\).

Therefore, the required probability is \(\frac{2}{3}\).

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Most popular questions from this chapter

Question: Prove Theorem \(2\), the extended form of Bayes’ theorem. That is, suppose that \(E\) is an event from a sample space \(S\) and that \({F_1},{F_2},...,{F_n}\) are mutually exclusive events such that \(\bigcup\nolimits_{i = 1}^n {{F_i} = S} \). Assume that \(p\left( E \right) \ne 0\) and \(p\left( {{F_i}} \right) \ne 0\) for \(i = 1,2,...,n\). Show that

\(p\left( {{F_j}\left| E \right.} \right) = \frac{{p\left( {E\left| {{F_j}} \right.} \right)p\left( {{F_j}} \right)}}{{\sum\nolimits_{i = 1}^n {p\left( {E\left| {{F_i}} \right.} \right)p\left( {{F_i}} \right)} }}\)

(Hint: use the fact that \(E = \bigcup\nolimits_{i = 1}^n {\left( {E \cap {F_i}} \right)} \).)

Question: What is the probability that a five-card poker hand does not contain the queen of hearts?

Question: A group of six people play the game of “odd person out” to determine who will buy refreshments. Each person flips a fair coin. If there is a person whose outcome is not the same as that of any other member of the group, this person has to buy the refreshments. What is the Probability that there is an odd person out after the coins are flipped once?

Question:Ramesh can get to work in three different ways: by bicycle, by car, or by bus. Because of commuter traffic, there is a\({\bf{50}}\% \)chance that he will be late when he drives his car. When he takes the bus, which uses a special lane reserved for buses, there is a\({\bf{20}}\% \)chance that he will be late. The probability that he is late when he rides his bicycle is only\({\bf{5}}\% \). Ramesh arrives late one day. His boss wants to estimate the probability that he drove his car to work that day.

a) Suppose the boss assumes that there is a\({\bf{1}}/{\bf{3}}\)chance that Ramesh takes each of the three ways he can get to work. What estimate for the probability that Ramesh drove his car does the boss obtain from Bayes’ theorem under this assumption?

b) Suppose the boss knows that Ramesh drives\(3{\bf{0}}\% \)of the time, takes the bus only\({\bf{10}}\)% of the time, and takes his bicycle\({\bf{60}}\% \)of the time. What estimate for the probability that Ramesh drove his car does the boss obtain from Bayes’ theorem using this information?

Question:To determine which is more likely: rolling a total of 9 when two dice are rolled or rolling a total of 9 when three dice are rolled?
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