Bishop or Knight: Which is more valuable?

Now you might have seen the title of this discussion and thought I was an idiot. After all, it's seems like pretty common knowledge that a bishop and a knight are of equal value. However, I raise this question because in the book Bobby Fischer Teaches Chess, Fischer writes that a bishop has a material value of 3.5 points while a knight has a value of only 3. This is not what I have heard elsewhere. Has the opinion changed on this matter? 

In some positions like, this the bishop is better in some positions like:

The bishop can maneuver in diagonal lines which are completely free.The knight,which is more slowly may be worse in the position i showed you above^^.In some positions the knight is better like this position:

In this position the bishop is COMPLETELY TRAPPED.So the knight is better.

But the question is WHICH IS MORE VALUABLE?Well i guess both are equal.Does this answer your question?I mean,I am just an intermediate but do you get it from this mini-lesson?Just give me the compliments in a private comment/message. 
 

I suggest simply bearing in mind the nominal 'value' of the pieces and trying to focus more on what each piece is doing. For example, if you have a situation where you are trying to decide whether to exchange your bishop for your opponents knight don't decide not to on the simple basis that the bishop is 'worth' slightly more. Look instead at what you judge will be the current and future role of both pieces in the game. Will your bishop be really useful/do you have a plan of how to use it well? If yes then perhaps don't exchange it. Is your opponents knight going to jump to places you don't want it to go and be really dangerous? Perhaps then exchange it. The more you play the better you will become at making these judgements. The same applies to exchange sacrifices eg.rook for bishop or pawn sacrifices. It is the position both current and potentially in the future (which is where you have to try to analyse) which is important rather than the +1 or -2 or whatever material count.

In the first example, I'd rather have white and the knight because the pawn at E5 is toast and the pawn at H6 looks suspect too, and whites pawns are well back making them harder to get before help arrives...white will have only to get the a6 pawn for at least a draw and it looks more difficult for black

The knight is a short ranged piece, but in closed positions, where the pawn center is locked, the knight is more powerful because he can jump around while other pieces can't. The bishop is more powerful in open positions, and where there are less pieces on the board, because two bishops can check mate while two knights can't force a mate unless a trade transposes directly into the mating position. Another consideration is that a bishop may be a long-ranged piece, it can only ever cover half the board. The bishop pair for this reason can be very powerful. This doesn't mean, however, to try to keep the bishop pair in closed positions. The bishop and knight are close enough in value that it all depends on the position. If your opponent has two rooks, and a queen in a position that a knight can penetrate, and create a lot of threats, with a great pawn structure to keep the knight in your opponent's territory, the knight is better. If the position looks like a lot of back-rank threats and pins, with a non-existent pawn center, the bishops are going to be better. There are even positions where a bishop or knight is more valuable than a rook, it all depends on the position and what kind of combinations that exist in it. Otherwise, generally, you want to avoid trades if you're playing for the win. Chess is a game of tennis, you hit the ball to the left, they return the ball, and you spike it to the right. When your opponent winds up on the defensive side, and you manage to create two threats at once, that's how to use your pieces. Especially with tied material, it's all about piece activity. Sometimes you need to sacrifice material just to get some piece activity, such as sacrificing a bishop for the pawn near opponent's king. Then you can go on for the draw with perpetual check using your queen, etc.

i have never come across the bishop thing and your question if they were both threatened to be taken they would both be as unsecure as the other your question is could one be more unsecure than the other? The answer is no 

Bishops are worth slightly more than Knights. This has been accepted since the days of Morphy. I don't know exactly how many centi-pawns Bs are worth compared to Knights ...

I just looked up Stockfish's generic evaluation of the pieces. It's a bit difficult to read, but here's about what Stockfish does:

Pawns are worth 1.008 pawns (they gain in value as they progress toward the 8th rank)

N= 3.191

B=3.266

R=4.961

Q=9.848

Source: http://www.talkchess.com/forum/viewtopic.php?topic_view=threads&p=442995&t=41916

There are some positions where Ns are worth more than Bs, but two Bs are worth about a pawn more than two Ns.  

The most important thing is to remember these values change with the position. Play the position and work out which piece is better in that given position. 

Fischer said a Bishop is worth 3.25, not 3.5.

Yes Bishop is worth something like 3.15-3.25 and because of this the bishop is slightly better over a knight. 

But this is not accurate as of course statistically it is better, but is it better positionally? What I mean is that the position determines if pieces are better or not.

A bishop would be better in open positions with doubledge play and pawns on both sides of the board.

However the knight would be significantly better in closed positions and better with same side maneuvering with pawns. 

It always comes down to the position. Chances are for the majority of the time, unless I had some 'brilliant" plan, I think that I would take the knight over the bishop in a closed position, and take a bishop over a knight in an open position that can have more play.

It all depends, there is no concrete answer on the board unless you determine the position, however stating from statistical point of view in theory the bishop is better. 

My opinion is that a knight is only good with another knight, like team work, while a bishop is good on it's own.

@harrisisawesome, let me know if you’d like to learn about chess. I offer lessons at a fair rate and my students generally improve dramatically.

Sometimes bishops are better because the have long ranges...

Sometimes knights are better because they are sneaky at times...

P.S. Isn't that second position in post #2 illegal?

Yes it is, @eric0022.

+1

In my opinion two bishops are always more deadlier than two knights.....

So in my opinion in most of the cases ....

For 3 pieces out of 4 : 2Bishops+Knight > 2Knights+Bishop

For 2 pieces out of 4 : 2Bishops > 1Bishop+1Knight > 2Knights

For 1 piece out of 4 : It depends. But mostly 1Knight>1Bishop

The fact that 2 Bishops and a King can deliver a checkmate whereas 2 Knights and a King can't is enough to prove that the Bishop is slightly superior to the knight.

In this game my bishop became useless and moveless at the end  due to blunders I made  (I'm black)

So it basically depends on position

Ya I blundered it for free, but I'm saying that sometimes bishop gets trapped like this.

Bishops tend to be superior when there is play on both wings, because a Bishop can influence play on both wings at once (eg: assisting the defense on one flank while supporting an attack on the other flank). Knights, being short-ranged pieces, cannot do this.

On the whole I'd say the Bishop edges it, both have strengths. However, the fact that the bishop has a larger range gives the Bishop a small advantage.